Introduction
The Alexandrov–Fenchel inequalities are fundamental inequalities in convex geometry. In Euclidean space, they give sharp comparisons between the quermassintegrals of a convex body, with equality precisely for balls. We refer to Schneider [31 ] for the classical theory. Curvature flows provide an important approach to these inequalities by deforming a hypersurface towards a sphere while preserving one geometric quantity and decreasing another. Important examples include the volume-preserving mean curvature flow of Huisken [19 ] , the mixed-volume-preserving curvature flows of McCoy [28 ] , and the inverse curvature flow of Guan and Li [14 ] . The latter also gives the quermassintegral inequalities for suitable k k k
For convex domains in hyperbolic space, the quermassintegrals are naturally defined by integral geometry. Unlike their Euclidean counterparts, their curvature integral expressions contain additional lower order terms arising from the ambient curvature. We refer to Santaló [30 ] and Solanes [32 ] for the corresponding integral geometric formulas.
Let H n + 1 \mathbb{H}^{n+1} H n + 1 − 1 -1 − 1 Ω ⊂ H n + 1 \Omega\subset\mathbb{H}^{n+1} Ω ⊂ H n + 1 M = ∂ Ω M = \partial\Omega M = ∂ Ω κ = ( κ 1 , … , κ n ) \kappa= (\kappa_1,\ldots,\kappa_n) κ = ( κ 1 , … , κ n ) ν \nu ν
E j ( κ ) = ( n j ) − 1 σ j ( κ ) , σ j ( κ ) = ∑ i 1 < ⋯ < i j κ i 1 ⋯ κ i j , E 0 = 1. E_j(\kappa) = \binom{n}{j}^{-1}\sigma_j(\kappa), \qquad\sigma_j(\kappa) = \sum_{i_1<\cdots<i_j}\kappa_{i_1}\cdots\kappa_{i_j}, \qquad E_0=1. E j ( κ ) = ( j n ) − 1 σ j ( κ ) , σ j ( κ ) = i 1 < ⋯ < i j ∑ κ i 1 ⋯ κ i j , E 0 = 1. Let W j W_j W j 0 ≤ j ≤ n 0 \le j \le n 0 ≤ j ≤ n [2 ] . Their definition and first variation are recalled in Section § 2.3 . For a geodesic ball B r B_r B r r > 0 r>0 r > 0
f j ( r ) = W j ( B r ) , 0 ≤ j ≤ n . f_j(r) = W_j(B_r), \qquad0 \le j \le n. f j ( r ) = W j ( B r ) , 0 ≤ j ≤ n . These functions are strictly increasing. The hyperbolic Alexandrov–Fenchel problem is to establish the sharp comparison
W j ( Ω ) ≥ f j ∘ f ℓ − 1 ( W ℓ ( Ω ) ) , 0 ≤ ℓ < j ≤ n , (1) W_j(\Omega) \ge f_j \circ f_\ell^{-1}(W_\ell(\Omega)), \qquad0 \le\ell< j \le n, \tag*{(1)} W j ( Ω ) ≥ f j ∘ f ℓ − 1 ( W ℓ ( Ω )) , 0 ≤ ℓ < j ≤ n , (1) with equality precisely for geodesic balls. Different positive normalizations of the individual quermassintegrals do not change this comparison. In particular, the normalization used in [17 ] is ( n + 1 − j ) / ( n + 1 ) (n+1-j)/(n+1) ( n + 1 − j ) / ( n + 1 ) W j W_j W j
A smooth convex hypersurface is called horospherically convex if κ i ≥ 1 \kappa_i \ge1 κ i ≥ 1 i i i [9 ] proved long-time existence and convergence of the volume-preserving mean curvature flow. Wang and Xia [35 ] established the full family of inequalities (1) by quermassintegral-preserving curvature flows. Andrews and Wei [6 ] subsequently proved convergence for a broad class of such flows. Hu, Li and Wei [18 ] gave another proof of the inequalities (1) using locally constrained curvature flows. Related sharp inequalities involving curvature integrals and quermassintegrals were obtained by Ge, Wang and Wu [11 ] .
Several cases are known under weaker geometric assumptions. Brendle, Guan and Li [7 ] proved the comparison between W 2 W_2 W 2 W 1 W_1 W 1 [15 , 18 ] . Li, Wei and Xiong [22 ] established the comparison between W 3 W_3 W 3 W 1 W_1 W 1 [16 ] extended the odd-index quermassintegral comparisons with W 1 W_1 W 1 [11 ] to hypersurfaces with nonnegative intrinsic sectional curvature. At the other end of the index range, Brendle, Guan and Li [7 ] proved (1) for smooth strictly convex domains with 0 ≤ ℓ < j = n 0 \le\ell< j = n 0 ≤ ℓ < j = n [4 ] used the harmonic mean curvature flow to establish several comparisons with upper index n − 1 n-1 n − 1 [17 ] established all comparisons with upper index n − 1 n-1 n − 1 [27 ] gave another proof of the comparison between W n − 1 W_{n-1} W n − 1 W 0 W_0 W 0
Of particular relevance to the present work is the volume-preserving flow of Andrews, Chen and Wei [2 ] . They proved long-time existence and exponential convergence to a geodesic sphere for a class of speeds including positive powers of the normalized mean curvatures, assuming that the initial hypersurface has positive intrinsic sectional curvature. For a convex boundary, the Gauss equation identifies this assumption with
κ i κ j > 1 for all i ≠ j . (2) \kappa_i\kappa_j > 1 \qquad\text{for all } i \ne j. \tag*{(2)} κ i κ j > 1 for all i = j . (2) This condition does not require each principal curvature to be at least one. Their convergence theorem yields
W j ( Ω ) ≥ f j ∘ f 0 − 1 ( W 0 ( Ω ) ) , 1 ≤ j ≤ n , (3) W_j(\Omega) \ge f_j \circ f_0^{-1}(W_0(\Omega)), \qquad1 \le j \le n, \tag*{(3)} W j ( Ω ) ≥ f j ∘ f 0 − 1 ( W 0 ( Ω )) , 1 ≤ j ≤ n , (3) for domains whose boundaries have positive sectional curvature, with equality precisely for geodesic balls. For the volume-preserving Gauss curvature flow, Wei, Yang and Zhou [36 ] subsequently proved long-time existence and smooth convergence for every positive power of the Gauss curvature under strict convexity alone. Their result also gives the comparison between W n W_n W n W 0 W_0 W 0
In this paper we consider W ℓ W_\ell W ℓ ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 [2 ] .
Main results
Our main geometric consequence is the following family of Alexandrov–Fenchel inequalities.
Theorem 1.1. Let Ω ⊂ H n + 1 \Omega\subset\mathbb{H}^{n+1} Ω ⊂ H n + 1 n ≥ 3 n \ge3 n ≥ 3 0 ≤ ℓ < k ≤ n 0 \le\ell< k \le n 0 ≤ ℓ < k ≤ n k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n
W k ( Ω ) ≥ f k ∘ f ℓ − 1 ( W ℓ ( Ω ) ) . (4) W_k(\Omega) \ge f_k \circ f_\ell^{-1}(W_\ell(\Omega)). \tag*{(4)} W k ( Ω ) ≥ f k ∘ f ℓ − 1 ( W ℓ ( Ω )) . (4) Equality holds if and only if Ω \Omega Ω
The comparisons with ℓ = 0 \ell=0 ℓ = 0 [2 ] . We include ℓ = 0 \ell=0 ℓ = 0 Theorem 1.1 to state the inequalities uniformly under nonnegative sectional curvature. For k ≤ ⌊ ( n + 1 ) / 2 ⌋ k \le\lfloor(n+1)/2 \rfloor k ≤ ⌊( n + 1 ) /2 ⌋ Theorem 1.1 gives all comparisons with 0 ≤ ℓ < k 0 \le\ell< k 0 ≤ ℓ < k n = 5 n=5 n = 5 W 3 W_3 W 3 W 2 W_2 W 2 n − 1 n-1 n − 1 n n n [17 ] and Brendle, Guan and Li [7 ] , respectively, we obtain the following result.
Corollary 1.2. Let 2 ≤ n ≤ 5 2 \le n \le5 2 ≤ n ≤ 5 Ω ⊂ H n + 1 \Omega\subset\mathbb{H}^{n+1} Ω ⊂ H n + 1
W k ( Ω ) ≥ f k ∘ f ℓ − 1 ( W ℓ ( Ω ) ) , 0 ≤ ℓ < k ≤ n . (5) W_k(\Omega) \ge f_k \circ f_\ell^{-1}(W_\ell(\Omega)), \qquad0 \le\ell< k \le n. \tag*{(5)} W k ( Ω ) ≥ f k ∘ f ℓ − 1 ( W ℓ ( Ω )) , 0 ≤ ℓ < k ≤ n . (5) Equality holds if and only if Ω \Omega Ω
Fix integers 1 ≤ ℓ < k ≤ n − 1 1 \le\ell< k \le n-1 1 ≤ ℓ < k ≤ n − 1 k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n X 0 X_0 X 0 M n M^n M n H n + 1 \mathbb{H}^{n+1} H n + 1
{ ∂ X ∂ t ( x , t ) = ( ϕ ( t ) − F ( W ) ) ν ( x , t ) , X ( ⋅ , 0 ) = X 0 , (6) \begin{cases}
\frac{\partial X}{\partial t}(x,t)=(\phi(t)-F(\mathcal{W}))\nu(x,t),\\
X(\cdot,0)=X_0, \tag*{(6)}
\end{cases} { ∂ t ∂ X ( x , t ) = ( ϕ ( t ) − F ( W )) ν ( x , t ) , X ( ⋅ , 0 ) = X 0 , (6) where
F = ( E k E ℓ ) 1 / ( k − ℓ ) , ϕ ( t ) = ∫ M t E ℓ F d μ t ∫ M t E ℓ d μ t . (7) F=\left(\frac{E_k}{E_\ell}\right)^{1/(k-\ell)}, \qquad\phi(t)=\frac{\int_{M_t}E_\ell F\,\mathrm{d}\mu_t}{\int_{M_t}E_\ell\,\mathrm{d}\mu_t}. \tag*{(7)} F = ( E ℓ E k ) 1/ ( k − ℓ ) , ϕ ( t ) = ∫ M t E ℓ d μ t ∫ M t E ℓ F d μ t . (7) Here M t = X ( M , t ) M_t=X(M,t) M t = X ( M , t ) W = ( h j i ) \mathcal{W}=(h^i_j) W = ( h j i ) ν \nu ν d μ t \mathrm{d}\mu_t d μ t
This flow belongs to the quermassintegral-preserving quotient family considered in [35 ] for horospherically convex hypersurfaces. It has the additive form studied in [2 ] , with the global term chosen to preserve W ℓ W_\ell W ℓ W ℓ W_\ell W ℓ W k W_k W k (30) , W k W_k W k F F F
Theorem 1.3. Let n ≥ 3 n \ge3 n ≥ 3 1 ≤ ℓ < k ≤ n − 1 1 \le\ell< k \le n-1 1 ≤ ℓ < k ≤ n − 1 k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n X 0 X_0 X 0 M n M^n M n H n + 1 \mathbb{H}^{n+1} H n + 1 M 0 = X 0 ( M ) M_0=X_0(M) M 0 = X 0 ( M ) Ω 0 \Omega_0 Ω 0 (6) –(7) has a unique smooth solution for all t ∈ [ 0 , ∞ ) t \in[0,\infty) t ∈ [ 0 , ∞ ) M t M_t M t M t M_t M t ∂ B r ∞ ( p ∞ ) \partial B_{r_\infty}(p_\infty) ∂ B r ∞ ( p ∞ ) W ℓ ( B r ∞ ) = W ℓ ( Ω 0 ) W_\ell(B_{r_\infty})=W_\ell(\Omega_0) W ℓ ( B r ∞ ) = W ℓ ( Ω 0 )
Since W ℓ W_{\ell} W ℓ W k W_k W k Theorem 1.3 proves (4) for 1 ≤ ℓ < k 1 \le\ell< k 1 ≤ ℓ < k k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n ℓ = 0 \ell=0 ℓ = 0 [2 ] . Klein contraction and a short-time flow argument extend the result, including the equality case, to nonnegative sectional curvature, as shown in § 4.4 .
The flow (1.6)–(1.7) is also defined for 1 ≤ ℓ < k ≤ n 1 \le\ell< k \le n 1 ≤ ℓ < k ≤ n k = n k=n k = n k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n
Theorem 1.4. Let n ≥ 3 n \ge3 n ≥ 3 1 ≤ ℓ < k ≤ n 1 \le\ell< k \le n 1 ≤ ℓ < k ≤ n k + ℓ > n k+\ell> n k + ℓ > n
Together with Theorem 1.3 , this gives the exact range k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n F = E m + 1 / E m F=E_{m+1}/E_m F = E m + 1 / E m 2 m < n 2m<n 2 m < n 2 m ≥ n 2m \ge n 2 m ≥ n
1.2. Discussion of the proof. We first prove preservation of positive sectional curvature for the quotient flow. Following [2 ] , we consider smooth lower supports of the minimum sectional curvature S = κ 1 κ 2 − 1 S=\kappa_1\kappa_2-1 S = κ 1 κ 2 − 1 F F F S S S
κ 2 ∇ m h 11 + κ 1 ∇ m h 22 = 0. \kappa_2\nabla_m h_{11}+\kappa_1\nabla_m h_{22}=0. κ 2 ∇ m h 11 + κ 1 ∇ m h 22 = 0. This relation allows us to apply the paired concavity inequality in Proposition 6.1 to the dual curvature quotient. The inequality holds precisely when k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n ϕ ( t ) \phi(t) ϕ ( t )
The main additional difficulty for the W ℓ W_{\ell} W ℓ [2 ] . Unlike the volume-preserving case, a positive lower bound for W ℓ W_{\ell} W ℓ ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 Lemma 4.2 , we use the sectional curvature condition and a three-dimensional section of a John ellipsoid to show that, if its shortest semiaxis tends to zero, then so does its second largest semiaxis. Thus the domains can only collapse to sets of dimension at most one, which would force W ℓ W_{\ell} W ℓ
With the inner and outer radius bounds, a Tso-type estimate [34 ] gives an upper bound for F F F − ϕ W 2 -\phi W^2 − ϕ W 2
The decrease of W k W_k W k E k / E ℓ E_k/E_{\ell} E k / E ℓ [21 ] . The linearization and normal stability theorem [29 ] then give smooth exponential convergence of the entire flow. For the Alexandrov–Fenchel inequalities, Klein contraction allows us to pass from positive to nonnegative sectional curvature. The equality cases follow by running the constrained flow for a short time and using the monotonicity of W k W_k W k
Finally, we show that the index restriction is necessary for curvature preservation by this family of flows. When k + ℓ > n k+\ell>n k + ℓ > n S = 0 S=0 S = 0
Organization of the paper
Section § 2 collects the curvature-function properties, hyperbolic geometry and Klein contraction, quermassintegral variations, and evolution equations used below. § 3 establishes preservation of positive sectional curvature. In § 4 , we prove the shape and curvature estimates, long-time existence, convergence, and the Alexandrov–Fenchel inequalities. § 5 proves sharpness of the index range by geometric counterexamples. The proof of the paired concavity inequality is given in § 6 .
On the use of AI
The main mathematical ideas and geometric strategy of this paper were developed by the authors. GPT-5.6 Sol and GPT-6 Astra were used to assist with calculations and proof development, particularly the algebraic inequality in § 6 , as well as with improving the exposition and correcting grammatical errors. The authors take full responsibility for the mathematical content and conclusions of this paper.
Preliminaries
In this section, we recall the properties of elementary symmetric functions and the geometry of hypersurfaces in hyperbolic space. We also record the Klein contraction, the variational formulas for quermassintegrals, and the evolution equations used below. Repeated tensor indices are summed from 1 to n n n
Elementary symmetric functions
We use the normalized elementary symmetric functions
E m ( κ ) = ( n m ) − 1 σ m ( κ ) , 0 ≤ m ≤ n . E_m(\kappa) = \binom{n}{m}^{-1}\sigma_m(\kappa), \qquad0 \le m \le n. E m ( κ ) = ( m n ) − 1 σ m ( κ ) , 0 ≤ m ≤ n . We set E 0 = 1 E_0=1 E 0 = 1 E m = 0 E_m=0 E m = 0 m < 0 m<0 m < 0 m > n m>n m > n
Γ + = { κ ∈ R n : κ i > 0 for all i } . \Gamma_+ = \{\kappa\in\mathbb{R}^n : \kappa_i>0 \text{ for all } i\}. Γ + = { κ ∈ R n : κ i > 0 for all i } . A smooth symmetric function Φ \Phi Φ Γ + \Gamma_+ Γ + A A A ξ \xi ξ
d d s Φ ( A + s ξ ) ∣ s = 0 = Φ ˙ i j ξ i j , d 2 d s 2 Φ ( A + s ξ ) ∣ s = 0 = Φ ¨ i j , k ℓ ξ i j ξ k ℓ . \left.\frac{d}{ds}\Phi(A+s\xi)\right|_{s=0}=\dot{\Phi}^{ij}\xi_{ij}, \qquad\left.\frac{d^2}{ds^2}\Phi(A+s\xi)\right|_{s=0}=\ddot{\Phi}^{ij,k\ell}\xi_{ij}\xi_{k\ell}. d s d Φ ( A + s ξ ) s = 0 = Φ ˙ ij ξ ij , d s 2 d 2 Φ ( A + s ξ ) s = 0 = Φ ¨ ij , k ℓ ξ ij ξ k ℓ . For derivatives in the eigenvalue variables, we write Φ ˙ i = ∂ Φ / ∂ κ i \dot{\Phi}^i=\partial\Phi/\partial\kappa_i Φ ˙ i = ∂ Φ/ ∂ κ i Φ ¨ i j = ∂ 2 Φ / ∂ κ i ∂ κ j \ddot{\Phi}^{ij}=\partial^2\Phi/\partial\kappa_i\partial\kappa_j Φ ¨ ij = ∂ 2 Φ/ ∂ κ i ∂ κ j [1 Theorem 5.1] gives
Φ ˙ i j = Φ ˙ i δ i j , \dot{\Phi}^{ij}=\dot{\Phi}^i\delta_{ij}, Φ ˙ ij = Φ ˙ i δ ij , Φ ¨ i j , k ℓ ξ i j ξ k ℓ = ∑ i , j Φ ¨ i j ξ i i ξ j j + 2 ∑ i < j Φ ˙ i − Φ ˙ j κ i − κ j ξ i j 2 . (8) \ddot{\Phi}^{ij,k\ell}\xi_{ij}\xi_{k\ell}
=
\sum_{i,j}\ddot{\Phi}^{ij}\xi_{ii}\xi_{jj}
+
2\sum_{i<j}\frac{\dot{\Phi}^i-\dot{\Phi}^j}{\kappa_i-\kappa_j}\xi_{ij}^2. \tag*{(8)} Φ ¨ ij , k ℓ ξ ij ξ k ℓ = i , j ∑ Φ ¨ ij ξ ii ξ j j + 2 i < j ∑ κ i − κ j Φ ˙ i − Φ ˙ j ξ ij 2 . (8) The divided differences extend continuously to repeated eigenvalues, where their value is Φ ¨ i i − Φ ¨ i j \ddot{\Phi}^{ii}-\ddot{\Phi}^{ij} Φ ¨ ii − Φ ¨ ij
On Γ + \Gamma_+ Γ + [18 Lemma 2.2] take the form
E m + 1 E j − 1 ≤ E m E j , 1 ≤ j ≤ m ≤ n − 1 , (9) E_{m+1}E_{j-1}\le E_mE_j, \qquad1\le j\le m\le n-1, \tag*{(9)} E m + 1 E j − 1 ≤ E m E j , 1 ≤ j ≤ m ≤ n − 1 , (9) with equality if and only if all principal curvatures are equal.
For 0 ≤ ℓ < k ≤ n 0\le\ell<k\le n 0 ≤ ℓ < k ≤ n
F = ( E k E ℓ ) 1 / ( k − ℓ ) . F=\left(\frac{E_k}{E_\ell}\right)^{1/(k-\ell)}. F = ( E ℓ E k ) 1/ ( k − ℓ ) . The adjacent quotients are strictly increasing and concave on Γ + \Gamma_+ Γ + [1 Theorem 2.6] . Since the geometric mean is increasing and concave, the representation
F = ( ∏ j = ℓ k − 1 E j + 1 E j ) 1 / ( k − ℓ ) . (10) F=\left(\prod_{j=\ell}^{k-1}\frac{E_{j+1}}{E_j}\right)^{1/(k-\ell)}. \tag*{(10)} F = j = ℓ ∏ k − 1 E j E j + 1 1/ ( k − ℓ ) . (10) shows that F F F
E m ( A − 1 ) = E n − m ( A ) E n ( A ) , 0 ≤ m ≤ n , E_m(A^{-1}) = \frac{E_{n-m}(A)}{E_n(A)}, \qquad0 \le m \le n, E m ( A − 1 ) = E n ( A ) E n − m ( A ) , 0 ≤ m ≤ n , and hence
F ∗ ( A ) : = 1 F ( A − 1 ) = ( E n − ℓ ( A ) E n − k ( A ) ) 1 / ( k − ℓ ) . (11) F_*(A) := \frac{1}{F(A^{-1})} = \left(\frac{E_{n-\ell}(A)}{E_{n-k}(A)}\right)^{1/(k-\ell)}. \tag*{(11)} F ∗ ( A ) := F ( A − 1 ) 1 = ( E n − k ( A ) E n − ℓ ( A ) ) 1/ ( k − ℓ ) . (11) The same argument with the complementary indices shows that F ∗ F_* F ∗ F F F
Homogeneity, monotonicity, and concavity give
F ˙ i j A i j = F , F ( Id ) = 1 , κ min ≤ F ≤ κ max , (12) \dot{F}^{ij} A_{ij} = F, \qquad F(\operatorname{Id}) = 1, \qquad\kappa_{\min} \le F \le\kappa_{\max}, \tag*{(12)} F ˙ ij A ij = F , F ( Id ) = 1 , κ m i n ≤ F ≤ κ m a x , (12) and
F ˙ i j δ i j ≥ 1 , F ˙ i j ( A 2 ) i j ≥ F 2 . (13) \dot{F}^{ij}\delta_{ij} \ge1, \qquad\dot{F}^{ij}(A^2)_{ij} \ge F^2. \tag*{(13)} F ˙ ij δ ij ≥ 1 , F ˙ ij ( A 2 ) ij ≥ F 2 . (13) The first inequality follows from concavity and Euler’s identity:
1 = F ( Id ) ≤ F ( A ) + F ˙ i j ( δ i j − A i j ) = F ˙ i j δ i j . 1 = F(\operatorname{Id}) \le F(A) + \dot{F}^{ij}(\delta_{ij} - A_{ij}) = \dot{F}^{ij}\delta_{ij}. 1 = F ( Id ) ≤ F ( A ) + F ˙ ij ( δ ij − A ij ) = F ˙ ij δ ij . Applying the same inequality to F ∗ F_* F ∗ A − 1 A^{-1} A − 1 F ˙ ∗ i ( κ − 1 ) = κ i 2 F ˙ i / F 2 \dot{F}_*^i(\kappa^{-1}) = \kappa_i^2\dot{F}^i/F^2 F ˙ ∗ i ( κ − 1 ) = κ i 2 F ˙ i / F 2 log F \log F log F [18 Lemma 2.1] , we obtain
F ˙ i j δ i j = F k − ℓ ( k E k − 1 E k − ℓ E ℓ − 1 E ℓ ) , \dot{F}^{ij}\delta_{ij} = \frac{F}{k-\ell}\left(k\frac{E_{k-1}}{E_k} - \ell\frac{E_{\ell-1}}{E_\ell}\right), F ˙ ij δ ij = k − ℓ F ( k E k E k − 1 − ℓ E ℓ E ℓ − 1 ) , F ˙ i j ( A 2 ) i j = F k − ℓ ( ( n − ℓ ) E ℓ + 1 E ℓ − ( n − k ) E k + 1 E k ) . (14) \dot{F}^{ij}(A^2)_{ij} = \frac{F}{k-\ell}\left((n-\ell)\frac{E_{\ell+1}}{E_\ell} - (n-k)\frac{E_{k+1}}{E_k}\right). \tag*{(14)} F ˙ ij ( A 2 ) ij = k − ℓ F ( ( n − ℓ ) E ℓ E ℓ + 1 − ( n − k ) E k E k + 1 ) . (14) For Section 3, put
ψ = − F − 1 , − ψ ( A − 1 ) = F ∗ ( A ) . (15) \psi= -F^{-1}, \qquad-\psi(A^{-1}) = F_*(A). \tag*{(15)} ψ = − F − 1 , − ψ ( A − 1 ) = F ∗ ( A ) . (15) Then ψ \psi ψ ψ \psi ψ F ∗ F_* F ∗
ψ ˙ i ≥ ψ ˙ j > 0 if κ i ≤ κ j , \dot{\psi}^i \ge\dot{\psi}^j > 0 \qquad\text{if } \kappa_i \le\kappa_j, ψ ˙ i ≥ ψ ˙ j > 0 if κ i ≤ κ j , κ i 2 ψ ˙ i ≤ κ j 2 ψ ˙ j if κ i ≤ κ j , \kappa_i^2\dot{\psi}^i \le\kappa_j^2\dot{\psi}^j \qquad\text{if } \kappa_i \le\kappa_j, κ i 2 ψ ˙ i ≤ κ j 2 ψ ˙ j if κ i ≤ κ j , ψ ˙ i − ψ ˙ j κ i − κ j + ψ ˙ i κ j + ψ ˙ j κ i ≥ 0 if i ≠ j . (16) \frac{\dot{\psi}^i-\dot{\psi}^j}{\kappa_i-\kappa_j}+\frac{\dot{\psi}^i}{\kappa_j}+\frac{\dot{\psi}^j}{\kappa_i}\ge0 \qquad\text{if } i\ne j. \tag*{(16)} κ i − κ j ψ ˙ i − ψ ˙ j + κ j ψ ˙ i + κ i ψ ˙ j ≥ 0 if i = j . (16) The last inequality follows from the second and F ˙ ∗ i ( κ − 1 ) = κ i 2 ψ ˙ i \dot{F}_*^i(\kappa^{-1}) = \kappa_i^2\dot{\psi}^i F ˙ ∗ i ( κ − 1 ) = κ i 2 ψ ˙ i
ψ ˙ i − ψ ˙ j κ i − κ j + ψ ˙ i κ j + ψ ˙ j κ i = κ i 2 ψ ˙ i − κ j 2 ψ ˙ j κ i κ j ( κ i − κ j ) . \frac{\dot{\psi}^i-\dot{\psi}^j}{\kappa_i-\kappa_j}+\frac{\dot{\psi}^i}{\kappa_j}+\frac{\dot{\psi}^j}{\kappa_i}
=\frac{\kappa_i^2\dot{\psi}^i-\kappa_j^2\dot{\psi}^j}{\kappa_i\kappa_j(\kappa_i-\kappa_j)}. κ i − κ j ψ ˙ i − ψ ˙ j + κ j ψ ˙ i + κ i ψ ˙ j = κ i κ j ( κ i − κ j ) κ i 2 ψ ˙ i − κ j 2 ψ ˙ j . At repeated eigenvalues, we use continuous extension.
Hypersurfaces in hyperbolic space
We use the standard conventions and identities for hypersurfaces in space forms as in [3 , 12 ] and [2 Section 2] . Fix p ∈ H n + 1 p \in\mathbb{H}^{n+1} p ∈ H n + 1 r r r p p p
g ˉ = d r 2 + sinh 2 r σ , \bar{g} = dr^2 + \sinh^2 r\,\sigma, g ˉ = d r 2 + sinh 2 r σ , where σ \sigma σ S n S^n S n ∇ ˉ \bar{\nabla} ∇ ˉ D D D g ˉ \bar{g} g ˉ σ \sigma σ
∇ ˉ U ( sinh r ∂ r ) = cosh r U . \bar{\nabla}_U(\sinh r\,\partial_r) = \cosh r\,U. ∇ ˉ U ( sinh r ∂ r ) = cosh r U . Let X X X M n M^n M n H n + 1 \mathbb{H}^{n+1} H n + 1 ν \nu ν g g g ∇ \nabla ∇ d μ d\mu d μ ∂ i = X ∗ ( ∂ / ∂ x i ) \partial_i = X_*(\partial/\partial x^i) ∂ i = X ∗ ( ∂ / ∂ x i )
g i j = g ˉ ( ∂ i , ∂ j ) , h i j = g ˉ ( ∇ ˉ ∂ i ν , ∂ j ) , W = ( h i j ) , h i j = g i k h k j . g_{ij} = \bar{g}(\partial_i,\partial_j), \qquad h_{ij} = \bar{g}(\bar{\nabla}_{\partial_i}\nu,\partial_j), \qquad\mathcal{W} = (h^i{}_j), \qquad h^i{}_j = g^{ik}h_{kj}. g ij = g ˉ ( ∂ i , ∂ j ) , h ij = g ˉ ( ∇ ˉ ∂ i ν , ∂ j ) , W = ( h i j ) , h i j = g ik h k j . The Weingarten map W \mathcal{W} W g g g κ 1 ≤ ⋯ ≤ κ n \kappa_1 \le\cdots\le\kappa_n κ 1 ≤ ⋯ ≤ κ n
H = tr W = n E 1 , ( h 2 ) i j = h i k g k ℓ h ℓ j . H = \operatorname{tr}\mathcal{W} = nE_1,\qquad(h^2)_{ij} = h_{ik}g^{k\ell}h_{\ell j}. H = tr W = n E 1 , ( h 2 ) ij = h ik g k ℓ h ℓ j . A geodesic sphere of radius r r r W = coth r Id \mathcal{W} = \coth r\,\operatorname{Id} W = coth r Id
R i j k ℓ = − ( g i k g j ℓ − g i ℓ g j k ) + h i k h j ℓ − h i ℓ h j k , (17) R_{ijk\ell} = -(g_{ik}g_{j\ell} - g_{i\ell}g_{jk}) + h_{ik}h_{j\ell} - h_{i\ell}h_{jk}, \tag*{(17)} R ij k ℓ = − ( g ik g j ℓ − g i ℓ g j k ) + h ik h j ℓ − h i ℓ h j k , (17) ∇ i h j k = ∇ j h i k . (18) \nabla_i h_{jk} = \nabla_j h_{ik}. \tag*{(18)} ∇ i h j k = ∇ j h ik . (18) In particular, a principal two-plane has sectional curvature κ i κ j − 1 \kappa_i\kappa_j - 1 κ i κ j − 1 ∇ i h j k \nabla_i h_{jk} ∇ i h j k M M M g − 1 / 2 h g − 1 / 2 g^{-1/2}hg^{-1/2} g − 1/2 h g − 1/2 (13) applies with δ i j \delta_{ij} δ ij ( A 2 ) i j (A^2)_{ij} ( A 2 ) ij g i j g_{ij} g ij ( h 2 ) i j (h^2)_{ij} ( h 2 ) ij
Suppose that M M M Ω \Omega Ω p ∈ int Ω p \in\operatorname{int}\Omega p ∈ int Ω
u = g ‾ ( sinh r ∂ r , ν ) . u = \overline{g}(\sinh r\,\partial_r,\nu). u = g ( sinh r ∂ r , ν ) . The support-function identities [18 Lemma 2.4] are
∇ i u = h i j ∇ j cosh r , \nabla_i u = h_i^{\ j}\nabla_j\cosh r, ∇ i u = h i j ∇ j cosh r , ∇ i ∇ j u = ∇ k cosh r ∇ k h i j + cosh r h i j − u ( h 2 ) i j . \nabla_i\nabla_j u = \nabla^k\cosh r\,\nabla_k h_{ij} + \cosh r\,h_{ij} - u(h^2)_{ij}. ∇ i ∇ j u = ∇ k cosh r ∇ k h ij + cosh r h ij − u ( h 2 ) ij . If B ρ ( p ) ⊂ Ω B_\rho(p) \subset\Omega B ρ ( p ) ⊂ Ω
u ≥ sinh ρ . (19) u \ge\sinh\rho. \tag*{(19)} u ≥ sinh ρ . (19) Writing M M M r = r ( θ ) r = r(\theta) r = r ( θ ) p p p θ ∈ S n \theta\in\mathbb{S}^n θ ∈ S n
r i = D i r , r i j = D i D j r , v = 1 + ∣ D r ∣ σ 2 sinh 2 r . r_i = D_i r,\qquad r_{ij} = D_iD_jr,\qquad v = \sqrt{1+\frac{|Dr|_\sigma^2}{\sinh^2 r}}. r i = D i r , r ij = D i D j r , v = 1 + sinh 2 r ∣ D r ∣ σ 2 . The radial graph identities [18 Section 2.3] are
g i j = sinh 2 r σ i j + r i r j , h i j = 1 v ( − r i j + sinh r cosh r σ i j + 2 coth r r i r j ) , u = sinh r v . (20) \begin{aligned}
g_{ij} &= \sinh^2 r\,\sigma_{ij} + r_i r_j,\\
h_{ij} &= \frac{1}{v}\left(-r_{ij} + \sinh r\cosh r\,\sigma_{ij} + 2\coth r\,r_i r_j\right),\\
u &= \frac{\sinh r}{v}. \tag*{(20)}
\end{aligned} g ij h ij u = sinh 2 r σ ij + r i r j , = v 1 ( − r ij + sinh r cosh r σ ij + 2 coth r r i r j ) , = v sinh r . (20) We also use the Klein model. In the hyperboloid realization, the inverse projection is
X = ( 1 , Y ) 1 − ∣ Y ∣ 2 , Y ∈ B 1 ( 0 ) ⊂ R n + 1 . (21) X = \frac{(1,Y)}{\sqrt{1-|Y|^2}},\qquad Y \in B_1(0) \subset\mathbb{R}^{n+1}. \tag*{(21)} X = 1 − ∣ Y ∣ 2 ( 1 , Y ) , Y ∈ B 1 ( 0 ) ⊂ R n + 1 . (21) Geodesic segments correspond to Euclidean line segments, so the Klein image of a convex domain is a Euclidean convex body. Let g E , h E g_E,h_E g E , h E B 1 ( 0 ) B_1(0) B 1 ( 0 ) ν E \nu_E ν E s = ⟨ Y , ν E ⟩ s = \langle Y,\nu_E\rangle s = ⟨ Y , ν E ⟩
ν H = ( s , ν E ) 1 − s 2 . \nu_H = \frac{(s,\nu_E)}{\sqrt{1-s^2}}. ν H = 1 − s 2 ( s , ν E ) . The pulled-back hyperbolic fundamental forms satisfy [6 Section 5]
( g H ) i j = 1 1 − ∣ Y ∣ 2 ( ( g E ) i j + ⟨ Y , Y i ⟩ ⟨ Y , Y j ⟩ 1 − ∣ Y ∣ 2 ) (22) (g_H)_{ij} = \frac{1}{1-|Y|^2}\left((g_E)_{ij} + \frac{\langle Y,Y_i\rangle\langle Y,Y_j\rangle}{1-|Y|^2}\right) \tag*{(22)} ( g H ) ij = 1 − ∣ Y ∣ 2 1 ( ( g E ) ij + 1 − ∣ Y ∣ 2 ⟨ Y , Y i ⟩ ⟨ Y , Y j ⟩ ) (22) and
h H = h E 1 − ∣ Y ∣ 2 1 − s 2 . (23) h_H = \frac{h_E}{\sqrt{1-|Y|^2}\sqrt{1-s^2}}. \tag*{(23)} h H = 1 − ∣ Y ∣ 2 1 − s 2 h E . (23) These formulas give the following strict approximation.
Lemma 2.1. Let M ⊂ H n + 1 M \subset\mathbb{H}^{n+1} M ⊂ H n + 1 n ≥ 2 n \ge2 n ≥ 2 Σ \Sigma Σ 0 < c ≤ 1 0 < c \le1 0 < c ≤ 1 M c M_c M c c Σ c\Sigma c Σ
κ i ( M c ) ≥ c − 1 κ i ( M ) , 1 ≤ i ≤ n . (24) \kappa_i(M_c) \ge c^{-1}\kappa_i(M), \qquad1 \le i \le n. \tag*{(24)} κ i ( M c ) ≥ c − 1 κ i ( M ) , 1 ≤ i ≤ n . (24) In particular, if sec M ≥ 0 \operatorname{sec}_M \ge0 sec M ≥ 0
sec M c ≥ c − 2 − 1 > 0 ( 0 < c < 1 ) , M c ⟶ M smoothly as c ↑ 1. (25) \operatorname{sec}_{M_c} \ge c^{-2} - 1 > 0 \quad(0 < c < 1), \qquad M_c \longrightarrow M \quad\text{smoothly as } c \uparrow1. \tag*{(25)} sec M c ≥ c − 2 − 1 > 0 ( 0 < c < 1 ) , M c ⟶ M smoothly as c ↑ 1. (25) Proof. Use the notation of (22) and (23) , with s = ⟨ Y , ν E ⟩ s = \langle Y,\nu_E\rangle s = ⟨ Y , ν E ⟩ Σ \Sigma Σ Y c = c Y Y_c = cY Y c = c Y
g E , c = c 2 g E , h E , c = c h E , ν E , c = ν E , s c = c s . g_{E,c} = c^2g_E, \qquad h_{E,c} = ch_E, \qquad\nu_{E,c} = \nu_E, \qquad s_c = cs. g E , c = c 2 g E , h E , c = c h E , ν E , c = ν E , s c = cs . Thus
g c ≤ c 2 ( 1 − ∣ Y ∣ 2 ) 1 − c 2 ∣ Y ∣ 2 g 1 , h c = c ( 1 − ∣ Y ∣ 2 ) ( 1 − s 2 ) ( 1 − c 2 ∣ Y ∣ 2 ) ( 1 − c 2 s 2 ) h 1 . (26) \begin{aligned}
g_c &\le\frac{c^2(1-\lvert Y\rvert^2)}{1-c^2\lvert Y\rvert^2}g_1,\\
h_c &= c\sqrt{\frac{(1-\lvert Y\rvert^2)(1-s^2)}{(1-c^2\lvert Y\rvert^2)(1-c^2s^2)}}h_1. \tag*{(26)}
\end{aligned} g c h c ≤ 1 − c 2 ∣ Y ∣ 2 c 2 ( 1 − ∣ Y ∣ 2 ) g 1 , = c ( 1 − c 2 ∣ Y ∣ 2 ) ( 1 − c 2 s 2 ) ( 1 − ∣ Y ∣ 2 ) ( 1 − s 2 ) h 1 . (26) The metric inequality follows from c 2 ( 1 − ∣ Y ∣ 2 ) ≤ 1 − c 2 ∣ Y ∣ 2 c^2(1-\lvert Y\rvert^2) \le1-c^2\lvert Y\rvert^2 c 2 ( 1 − ∣ Y ∣ 2 ) ≤ 1 − c 2 ∣ Y ∣ 2 h 1 > 0 h_1 > 0 h 1 > 0 V ≠ 0 V \ne0 V = 0
h c ( V , V ) g c ( V , V ) ≥ 1 c ( 1 − c 2 ∣ Y ∣ 2 ) ( 1 − s 2 ) ( 1 − ∣ Y ∣ 2 ) ( 1 − c 2 s 2 ) h 1 ( V , V ) g 1 ( V , V ) ≥ 1 c h 1 ( V , V ) g 1 ( V , V ) , \frac{h_c(V,V)}{g_c(V,V)} \ge\frac{1}{c}\sqrt{\frac{(1-c^2\lvert Y\rvert^2)(1-s^2)}{(1-\lvert Y\rvert^2)(1-c^2s^2)}}\frac{h_1(V,V)}{g_1(V,V)} \ge\frac{1}{c}\frac{h_1(V,V)}{g_1(V,V)}, g c ( V , V ) h c ( V , V ) ≥ c 1 ( 1 − ∣ Y ∣ 2 ) ( 1 − c 2 s 2 ) ( 1 − c 2 ∣ Y ∣ 2 ) ( 1 − s 2 ) g 1 ( V , V ) h 1 ( V , V ) ≥ c 1 g 1 ( V , V ) h 1 ( V , V ) , where the last inequality uses
( 1 − c 2 ∣ Y ∣ 2 ) ( 1 − s 2 ) − ( 1 − ∣ Y ∣ 2 ) ( 1 − c 2 s 2 ) = ( 1 − c 2 ) ( ∣ Y ∣ 2 − s 2 ) ≥ 0. (1-c^2\lvert Y\rvert^2)(1-s^2) - (1-\lvert Y\rvert^2)(1-c^2s^2) = (1-c^2)(\lvert Y\rvert^2-s^2) \ge0. ( 1 − c 2 ∣ Y ∣ 2 ) ( 1 − s 2 ) − ( 1 − ∣ Y ∣ 2 ) ( 1 − c 2 s 2 ) = ( 1 − c 2 ) (∣ Y ∣ 2 − s 2 ) ≥ 0. The min–max principle proves (24) . Equation (17) and smoothness of the inverse Klein map give (2.18)。
Hyperbolic quermassintegrals
For a smooth bounded convex domain Ω ⊂ H n + 1 \Omega\subset\mathbb{H}^{n+1} Ω ⊂ H n + 1 [2 (1.10)–(1.11)] :
W 0 ( Ω ) = ∣ Ω ∣ , W 1 ( Ω ) = ∣ ∂ Ω ∣ n , W_0(\Omega) = \lvert\Omega\rvert, \qquad W_1(\Omega) = \frac{\lvert\partial\Omega\rvert}{n}, W 0 ( Ω ) = ∣ Ω ∣ , W 1 ( Ω ) = n ∣ ∂ Ω ∣ , ( n − j ) W j + 1 ( Ω ) = ∫ ∂ Ω E j d μ − j W j − 1 ( Ω ) , 1 ≤ j ≤ n − 1. (27) (n-j)W_{j+1}(\Omega) = \int_{\partial\Omega} E_j\,d\mu- jW_{j-1}(\Omega), \qquad1 \le j \le n-1. \tag*{(27)} ( n − j ) W j + 1 ( Ω ) = ∫ ∂ Ω E j d μ − j W j − 1 ( Ω ) , 1 ≤ j ≤ n − 1. (27) The integral-geometric definition implies positivity, inclusion monotonicity, and continuity under Hausdorff convergence of convex bodies. We refer to [30 , 32 ] and [2 Section 1.2] . Under a normal variation ∂ t X = φ ν \partial_tX = \varphi\nu ∂ t X = φ ν [2 Lemma 2.3] is
d d t W j ( Ω t ) = ∫ M t E j φ d μ t , 0 ≤ j ≤ n . (28) \frac{d}{dt}W_j(\Omega_t) = \int_{M_t} E_j\varphi\,d\mu_t, \qquad0 \le j \le n. \tag*{(28)} d t d W j ( Ω t ) = ∫ M t E j φ d μ t , 0 ≤ j ≤ n . (28) For f j ( r ) = W j ( B r ) f_j(r) = W_j(B_r) f j ( r ) = W j ( B r )
f j ′ ( r ) = ω n sinh n − j r cosh j r > 0 , f j ( 0 ) = 0 , lim r → ∞ f j ( r ) = ∞ , (29) f_j'(r) = \omega_n\sinh^{n-j}r\cosh^j r > 0, \qquad f_j(0) = 0, \qquad\lim_{r\to\infty}f_j(r) = \infty, \tag*{(29)} f j ′ ( r ) = ω n sinh n − j r cosh j r > 0 , f j ( 0 ) = 0 , r → ∞ lim f j ( r ) = ∞ , (29) where ω n = ∣ S n ∣ \omega_n = \lvert S^n\rvert ω n = ∣ S n ∣ 0 ≤ j ≤ n 0 \le j \le n 0 ≤ j ≤ n f j − 1 f_j^{-1} f j − 1 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) (6) –(7) , we have
E k = E ℓ F k − ℓ , ∫ M t E ℓ ( F − φ ) d μ t = 0. E_k = E_\ell F^{k-\ell}, \qquad\int_{M_t} E_\ell(F-\varphi)\,d\mu_t = 0. E k = E ℓ F k − ℓ , ∫ M t E ℓ ( F − φ ) d μ t = 0. Substituting in (28) yields
d d t W ℓ ( Ω t ) = 0 , d d t W k ( Ω t ) = ∫ M t E ℓ F k − ℓ ( ϕ − F ) d μ t = − ∫ M t E ℓ ( F k − ℓ − ϕ k − ℓ ) ( F − ϕ ) d μ t ≤ 0. (30) \begin{aligned}
\frac{d}{dt}W_{\ell}(\Omega_t)&=0,\\
\frac{d}{dt}W_k(\Omega_t)&=\int_{M_t}E_{\ell}F^{k-\ell}(\phi-F)\,d\mu_t\\
&=-\int_{M_t}E_{\ell}(F^{k-\ell}-\phi^{k-\ell})(F-\phi)\,d\mu_t\leq0. \tag*{(30)}
\end{aligned} d t d W ℓ ( Ω t ) d t d W k ( Ω t ) = 0 , = ∫ M t E ℓ F k − ℓ ( ϕ − F ) d μ t = − ∫ M t E ℓ ( F k − ℓ − ϕ k − ℓ ) ( F − ϕ ) d μ t ≤ 0. (30) For a strictly convex hypersurface, the last integral vanishes if and only if F F F ℓ = 0 \ell=0 ℓ = 0 E 0 = 1 E_0=1 E 0 = 1 F = E k 1 / k F=E_k^{1/k} F = E k 1/ k
Evolution equations
We write the flow (6) as
∂ t X = − G ν , G = F − ϕ ( t ) , F = ( E k E ℓ ) 1 / ( k − ℓ ) . (31) \partial_tX=-G\nu,\qquad G=F-\phi(t),\qquad F=\left(\frac{E_k}{E_\ell}\right)^{1/(k-\ell)}. \tag*{(31)} ∂ t X = − G ν , G = F − ϕ ( t ) , F = ( E ℓ E k ) 1/ ( k − ℓ ) . (31) The global term ϕ ( t ) \phi(t) ϕ ( t )
G ˙ i j = F ˙ i j , G ¨ i j , k l = F ¨ i j , k l . (32) \dot{G}^{ij}=\dot{F}^{ij},\qquad\ddot{G}^{ij,kl}=\ddot{F}^{ij,kl}. \tag*{(32)} G ˙ ij = F ˙ ij , G ¨ ij , k l = F ¨ ij , k l . (32) Set
L = G ˙ i j ∇ i ∇ j = F ˙ i j ∇ i ∇ j . (33) \mathcal{L}=\dot{G}^{ij}\nabla_i\nabla_j=\dot{F}^{ij}\nabla_i\nabla_j. \tag*{(33)} L = G ˙ ij ∇ i ∇ j = F ˙ ij ∇ i ∇ j . (33) The following evolution equations follow from [2 ] , (14) –(2.9), (2.12)–{xref}(2.13), and (68) , using F ˙ i j h i j = F \dot{F}^{ij}h_{ij}=F F ˙ ij h ij = F ϕ ( t ) \phi(t) ϕ ( t )
Lemma 2.2. Along (31) , the metric and area element satisfy
∂ t g i j = − 2 G h i j , ∂ t g i j = 2 G h i j , ∂ t ( d μ t ) = − G H d μ t . (34) \partial_tg_{ij}=-2Gh_{ij},\qquad\partial_tg^{ij}=2Gh^{ij},\qquad\partial_t(d\mu_t)=-GH\,d\mu_t. \tag*{(34)} ∂ t g ij = − 2 G h ij , ∂ t g ij = 2 G h ij , ∂ t ( d μ t ) = − G H d μ t . (34) In local coordinates, the second fundamental form satisfies
∂ t h i j = ∇ i ∇ j G − G ( ( h 2 ) i j + g i j ) (35) \partial_th_{ij}=\nabla_i\nabla_jG-G\left((h^2)_{ij}+g_{ij}\right) \tag*{(35)} ∂ t h ij = ∇ i ∇ j G − G ( ( h 2 ) ij + g ij ) (35) and
∂ t h i j = F ˙ k ℓ ∇ k ∇ ℓ h i j + F ¨ k ℓ , p q ∇ i h k ℓ ∇ j h p q + F ˙ k ℓ ( ( h 2 ) k ℓ + g k ℓ ) h i j − ( 2 F − ϕ ) ( ( h 2 ) i j + g i j ) . (36) \begin{aligned}
\partial_th_{ij}&=\dot{F}^{k\ell}\nabla_k\nabla_\ell h_{ij}+\ddot{F}^{k\ell,pq}\nabla_i h_{k\ell}\nabla_jh_{pq}\\
&\quad+\dot{F}^{k\ell}\left((h^2)_{k\ell}+g_{k\ell}\right)h_{ij}-(2F-\phi)\left((h^2)_{ij}+g_{ij}\right). \tag*{(36)}
\end{aligned} ∂ t h ij = F ˙ k ℓ ∇ k ∇ ℓ h ij + F ¨ k ℓ , pq ∇ i h k ℓ ∇ j h pq + F ˙ k ℓ ( ( h 2 ) k ℓ + g k ℓ ) h ij − ( 2 F − ϕ ) ( ( h 2 ) ij + g ij ) . (36) The Weingarten map satisfies
( ∂ t − L ) h j i = F ¨ k ℓ , p q ∇ i h k ℓ ∇ j h p q + F ˙ k ℓ ( ( h 2 ) k ℓ + g k ℓ ) h j i − ϕ h k i h j k + ( ϕ − 2 F ) δ j i . (37) \begin{aligned}
(\partial_t-\mathcal{L})h^i_j&=\ddot{F}^{k\ell,pq}\nabla_i h_{k\ell}\nabla_jh_{pq}\\
&\quad+\dot{F}^{k\ell}\left((h^2)_{k\ell}+g_{k\ell}\right)h^i_j-\phi h^i_kh^k_j+(\phi-2F)\delta^i_j. \tag*{(37)}
\end{aligned} ( ∂ t − L ) h j i = F ¨ k ℓ , pq ∇ i h k ℓ ∇ j h pq + F ˙ k ℓ ( ( h 2 ) k ℓ + g k ℓ ) h j i − ϕ h k i h j k + ( ϕ − 2 F ) δ j i . (37) Moreover,
( ∂ t − L ) F = ( F − ϕ ) F ˙ i j ( ( h 2 ) i j − g i j ) , (38) (\partial_t-\mathcal{L})F=(F-\phi)\dot{F}^{ij}\left((h^2)_{ij}-g_{ij}\right), \tag*{(38)} ( ∂ t − L ) F = ( F − ϕ ) F ˙ ij ( ( h 2 ) ij − g ij ) , (38) ( ∂ t − L ) F − 1 = − 2 F 3 F ˙ i j ∇ i F ∇ j F − F − ϕ F 2 F ˙ i j ( ( h 2 ) i j − g i j ) , (39) (\partial_t-\mathcal{L})F^{-1}=-\frac{2}{F^3}\dot{F}^{ij}\nabla_iF\nabla_jF-\frac{F-\phi}{F^2}\dot{F}^{ij}\left((h^2)_{ij}-g_{ij}\right), \tag*{(39)} ( ∂ t − L ) F − 1 = − F 3 2 F ˙ ij ∇ i F ∇ j F − F 2 F − ϕ F ˙ ij ( ( h 2 ) ij − g ij ) , (39) and the support function about a fixed point satisfies
( ∂ t − L ) u = ( ϕ − 2 F ) cosh r + u F ˙ i j ( h 2 ) i j . (40) (\partial_t-\mathcal{L})u=(\phi-2F)\cosh r+u\dot{F}^{ij}(h^2)_{ij}. \tag*{(40)} ( ∂ t − L ) u = ( ϕ − 2 F ) cosh r + u F ˙ ij ( h 2 ) ij . (40) For every 0 ≤ ℓ < k ≤ n 0\leq\ell<k\leq n 0 ≤ ℓ < k ≤ n (6) –(7) is uniformly parabolic near a smooth closed strictly convex hypersurface. Here ℓ = 0 \ell=0 ℓ = 0 F = E k 1 / k F=E_k^{1/k} F = E k 1/ k ϕ = ∣ M t ∣ − 1 ∫ M t F d μ t \phi=|M_t|^{-1}\int_{M_t}F\,d\mu_t ϕ = ∣ M t ∣ − 1 ∫ M t F d μ t ϕ \phi ϕ h 2 + γ h^{2+\gamma} h 2 + γ h 4 + γ h^{4+\gamma} h 4 + γ 0 < γ < 1 0<\gamma<1 0 < γ < 1 [25 ] , Sections 8.1 and 8.3 therefore gives a unique smooth short-time solution. Nearby smooth initial hypersurfaces have solutions on a common time interval, depending continuously on the initial data in C 2 C^2 C 2 [ 0 , T max ) [0,T_{\max}) [ 0 , T m a x )
Preservation of positive sectional curvature
We prove preservation of positive sectional curvature for any positive smooth function ϕ ( t ) \phi(t) ϕ ( t )
Theorem 3.1. Let n ≥ 3 n \ge3 n ≥ 3 1 ≤ ℓ < k ≤ n − 1 1 \le\ell< k \le n-1 1 ≤ ℓ < k ≤ n − 1 k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n M t ⊂ H n + 1 M_t \subset\mathbb{H}^{n+1} M t ⊂ H n + 1
∂ t X = ( ϕ ( t ) − F ) ν , F = ( E k E ℓ ) 1 / ( k − ℓ ) , \partial_t X = (\phi(t)-F)\nu,\qquad F=\left(\frac{E_k}{E_\ell}\right)^{1/(k-\ell)}, ∂ t X = ( ϕ ( t ) − F ) ν , F = ( E ℓ E k ) 1/ ( k − ℓ ) , on [ 0 , T ) [0,T) [ 0 , T ) Γ + \Gamma_+ Γ + ϕ \phi ϕ M 0 M_0 M 0 M t M_t M t t ∈ [ 0 , T ) t \in[0,T) t ∈ [ 0 , T )
We use G = F − ϕ ( t ) G=F-\phi(t) G = F − ϕ ( t ) L = F ˙ i j ∇ i ∇ j \mathcal{L}=\dot{F}^{ij}\nabla_i\nabla_j L = F ˙ ij ∇ i ∇ j § 2.4 . Order the principal curvatures as 0 < κ 1 ≤ ⋯ ≤ κ n 0<\kappa_1\le\cdots\le\kappa_n 0 < κ 1 ≤ ⋯ ≤ κ n
S = κ 1 κ 2 − 1. S=\kappa_1\kappa_2-1. S = κ 1 κ 2 − 1. By the Gauss equation, S S S S S S
The function S S S [8 ] . A smooth function φ \varphi φ S S S ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) φ ≤ S \varphi\le S φ ≤ S φ ( x 0 , t 0 ) = S ( x 0 , t 0 ) \varphi(x_0,t_0)=S(x_0,t_0) φ ( x 0 , t 0 ) = S ( x 0 , t 0 ) φ \varphi φ t ≤ t 0 t\le t_0 t ≤ t 0
Lemma 3.2. Let φ \varphi φ S S S ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) t 0 > 0 t_0>0 t 0 > 0 g ( t 0 ) g(t_0) g ( t 0 ) { e 1 , … , e n } \{e_1,\ldots,e_n\} { e 1 , … , e n } x 0 x_0 x 0
J i = { p > 2 ∣ κ p > κ i } , i = 1 , 2. J_i=\{p>2\mid\kappa_p>\kappa_i\},\qquad i=1,2. J i = { p > 2 ∣ κ p > κ i } , i = 1 , 2. For a smooth symmetric curvature function Θ \Theta Θ
Q [ Θ ] = κ 2 Θ ¨ i j , p q ∇ 1 h i j ∇ 1 h p q + κ 1 Θ ¨ i j , p q ∇ 2 h i j ∇ 2 h p q − 2 ∑ m = 1 n Θ ˙ m ( ∇ m h 11 ∇ m h 22 − ( ∇ m h 12 ) 2 ) + 2 ∑ m = 1 n Θ ˙ m ∑ i = 1 2 ∑ p ∈ J i κ 1 κ 2 κ i ( κ p − κ i ) ( ∇ m h i p ) 2 . (41) \begin{aligned}
Q[\Theta] ={}& \kappa_2\ddot{\Theta}^{ij,pq}\nabla_1h_{ij}\nabla_1h_{pq}
+\kappa_1\ddot{\Theta}^{ij,pq}\nabla_2h_{ij}\nabla_2h_{pq}\\
&-2\sum_{m=1}^{n}\dot{\Theta}^{m}\left(\nabla_mh_{11}\nabla_mh_{22}-(\nabla_mh_{12})^2\right)\\
&+2\sum_{m=1}^{n}\dot{\Theta}^{m}\sum_{i=1}^{2}\sum_{p\in J_i}
\frac{\kappa_1\kappa_2}{\kappa_i(\kappa_p-\kappa_i)}
(\nabla_mh_{ip})^2. \tag*{(41)}
\end{aligned} Q [ Θ ] = κ 2 Θ ¨ ij , pq ∇ 1 h ij ∇ 1 h pq + κ 1 Θ ¨ ij , pq ∇ 2 h ij ∇ 2 h pq − 2 m = 1 ∑ n Θ ˙ m ( ∇ m h 11 ∇ m h 22 − ( ∇ m h 12 ) 2 ) + 2 m = 1 ∑ n Θ ˙ m i = 1 ∑ 2 p ∈ J i ∑ κ i ( κ p − κ i ) κ 1 κ 2 ( ∇ m h i p ) 2 . (41) Then, for m = 1 , … , n m=1,\ldots,n m = 1 , … , n
κ 2 ∇ m h 11 + κ 1 ∇ m h 22 = ∇ m φ , (42) \kappa_2\nabla_mh_{11}+\kappa_1\nabla_mh_{22}=\nabla_m\varphi, \tag*{(42)} κ 2 ∇ m h 11 + κ 1 ∇ m h 22 = ∇ m φ , (42) and
∇ m h i p = 0 if i ∈ { 1 , 2 } , p > 2 , κ p = κ i . (43) \nabla_mh_{ip}=0\qquad\text{if }i\in\{1,2\},\ p>2,\ \kappa_p=\kappa_i. \tag*{(43)} ∇ m h i p = 0 if i ∈ { 1 , 2 } , p > 2 , κ p = κ i . (43) Moreover, at the contact point,
( ∂ t − L ) φ ≥ Q [ G ] + R , (44) (\partial_t-\mathcal{L})\varphi\ge Q[G]+\mathcal{R}, \tag*{(44)} ( ∂ t − L ) φ ≥ Q [ G ] + R , (44) where
R = 2 F ˙ i j ( ( h 2 ) i j + g i j ) κ 1 κ 2 − ϕ κ 1 κ 2 ( κ 1 + κ 2 ) + ( ϕ − 2 F ) ( κ 1 + κ 2 ) . (45) \mathcal{R}=2\dot{F}^{ij}\left((h^2)_{ij}+g_{ij}\right)\kappa_1\kappa_2-\phi\kappa_1\kappa_2(\kappa_1+\kappa_2)+(\phi-2F)(\kappa_1+\kappa_2). \tag*{(45)} R = 2 F ˙ ij ( ( h 2 ) ij + g ij ) κ 1 κ 2 − ϕ κ 1 κ 2 ( κ 1 + κ 2 ) + ( ϕ − 2 F ) ( κ 1 + κ 2 ) . (45) Proof. All tensor components and derivatives below are evaluated at ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) w 1 , w 2 w_1,w_2 w 1 , w 2
det ( h ( w i , w j ) ) i , j = 1 2 = ∑ a < b κ a κ b ( w 1 a w 2 b − w 1 b w 2 a ) 2 ≥ κ 1 κ 2 , \det\bigl(h(w_i,w_j)\bigr)_{i,j=1}^{2}
=\sum_{a<b}\kappa_a\kappa_b\left(w_1^aw_2^b-w_1^bw_2^a\right)^2\ge\kappa_1\kappa_2, det ( h ( w i , w j ) ) i , j = 1 2 = a < b ∑ κ a κ b ( w 1 a w 2 b − w 1 b w 2 a ) 2 ≥ κ 1 κ 2 , since the squared minors sum to one. Equality holds for ( w 1 , w 2 ) = ( e 1 , e 2 ) (w_1,w_2)=(e_1,e_2) ( w 1 , w 2 ) = ( e 1 , e 2 )
f = det ( h ( w i , w j ) ) i , j = 1 2 − 1 f=\det\bigl(h(w_i,w_j)\bigr)_{i,j=1}^{2}-1 f = det ( h ( w i , w j ) ) i , j = 1 2 − 1 satisfying f ≥ S ≥ φ f \ge S \ge\varphi f ≥ S ≥ φ f − φ f-\varphi f − φ (42) .
To prove (43) , fix i ∈ { 1 , 2 } i \in\{1,2\} i ∈ { 1 , 2 } p > 2 p>2 p > 2 κ p = κ i \kappa_p=\kappa_i κ p = κ i j j j { 1 , 2 } \{1,2\} { 1 , 2 }
v ± = e i ± e p 2 . v_{\pm}=\frac{e_i\pm e_p}{\sqrt{2}}. v ± = 2 e i ± e p . Both orthonormal pairs ( v + , e j ) (v_+,e_j) ( v + , e j ) ( v − , e j ) (v_-,e_j) ( v − , e j ) κ 1 κ 2 \kappa_1\kappa_2 κ 1 κ 2
f ± = h ( v ± , v ± ) h ( e j , e j ) − h ( v ± , e j ) 2 − 1. f_{\pm}=h(v_{\pm},v_{\pm})h(e_j,e_j)-h(v_{\pm},e_j)^2-1. f ± = h ( v ± , v ± ) h ( e j , e j ) − h ( v ± , e j ) 2 − 1. Then f ± ≥ S ≥ φ f_{\pm}\ge S\ge\varphi f ± ≥ S ≥ φ ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) ∇ m f + = ∇ m f − = ∇ m φ \nabla_m f_+=\nabla_m f_-=\nabla_m\varphi ∇ m f + = ∇ m f − = ∇ m φ
0 = ∇ m f + − ∇ m f − = 2 κ j ∇ m h i p . 0=\nabla_m f_+-\nabla_m f_-=2\kappa_j\nabla_m h_{ip}. 0 = ∇ m f + − ∇ m f − = 2 κ j ∇ m h i p . Since κ j > 0 \kappa_j>0 κ j > 0 (43) .
We now choose the extension used in the evolution calculation. Extend the principal frame to a smooth local g ( t ) g(t) g ( t ) v 1 , … , v n v_1,\ldots,v_n v 1 , … , v n v i = e i v_i=e_i v i = e i
∇ m v i = − ∑ p ∈ J i ∇ m h i p κ p − κ i e p , i = 1 , 2. (46) \nabla_m v_i=-\sum_{p\in J_i}\frac{\nabla_m h_{ip}}{\kappa_p-\kappa_i}e_p,\qquad i=1,2. \tag*{(46)} ∇ m v i = − p ∈ J i ∑ κ p − κ i ∇ m h i p e p , i = 1 , 2. (46) Complete these spatial derivatives skew-symmetrically, with no rotation inside span { e 1 , e 2 } \operatorname{span}\{e_1,e_2\} span { e 1 , e 2 } (46) . The derivatives corresponding to zero gaps vanish by (43) . Empty sums are understood as zero. In the time direction, choose
∂ t v i = G W ( v i ) \partial_t v_i=GW(v_i) ∂ t v i = G W ( v i ) at the contact point, consistently with ∂ t g = − 2 G h \partial_t g=-2Gh ∂ t g = − 2 G h
For this frame, set
f = det ( h ( v i , v j ) ) i , j = 1 2 − 1. f=\det\bigl(h(v_i,v_j)\bigr)_{i,j=1}^{2}-1. f = det ( h ( v i , v j ) ) i , j = 1 2 − 1. At the contact point,
∇ m ( h ( v i , v j ) ) = ∇ m h i j , i , j ≤ 2. \nabla_m\bigl(h(v_i,v_j)\bigr)=\nabla_m h_{ij},\qquad i,j\le2. ∇ m ( h ( v i , v j ) ) = ∇ m h ij , i , j ≤ 2. Extend the direction e m e_m e m
⟨ ∇ m ∇ m v i , v i ⟩ = − ∣ ∇ m v i ∣ 2 . \langle\nabla_m\nabla_m v_i,v_i\rangle=-\lvert\nabla_m v_i\rvert^2. ⟨ ∇ m ∇ m v i , v i ⟩ = − ∣ ∇ m v i ∣ 2 . Together with (46) , this yields
∇ m ∇ m ( h ( v i , v i ) ) = ∇ m ∇ m h i i − 2 ∑ p ∈ J i ( ∇ m h i p ) 2 κ p − κ i , i = 1 , 2. \nabla_m\nabla_m\bigl(h(v_i,v_i)\bigr)=\nabla_m\nabla_m h_{ii}-2\sum_{p\in J_i}\frac{(\nabla_m h_{ip})^2}{\kappa_p-\kappa_i},\qquad i=1,2. ∇ m ∇ m ( h ( v i , v i ) ) = ∇ m ∇ m h ii − 2 p ∈ J i ∑ κ p − κ i ( ∇ m h i p ) 2 , i = 1 , 2. The second derivative formula for the two-dimensional determinant therefore gives
∇ m ∇ m f = κ 2 ∇ m ∇ m h 11 + κ 1 ∇ m ∇ m h 22 − 2 ∑ i = 1 2 ∑ p ∈ J i κ 1 κ 2 κ i ( κ p − κ i ) ( ∇ m h i p ) 2 + 2 ( ∇ m h 11 ∇ m h 22 − ( ∇ m h 12 ) 2 ) . (47) \begin{aligned}
\nabla_m\nabla_m f={}&\kappa_2\nabla_m\nabla_m h_{11}+\kappa_1\nabla_m\nabla_m h_{22}
-2\sum_{i=1}^{2}\sum_{p\in J_i}\frac{\kappa_1\kappa_2}{\kappa_i(\kappa_p-\kappa_i)}(\nabla_m h_{ip})^2\\
&+2\left(\nabla_m h_{11}\nabla_m h_{22}-(\nabla_m h_{12})^2\right). \tag*{(47)}
\end{aligned} ∇ m ∇ m f = κ 2 ∇ m ∇ m h 11 + κ 1 ∇ m ∇ m h 22 − 2 i = 1 ∑ 2 p ∈ J i ∑ κ i ( κ p − κ i ) κ 1 κ 2 ( ∇ m h i p ) 2 + 2 ( ∇ m h 11 ∇ m h 22 − ( ∇ m h 12 ) 2 ) . (47) For the time derivative, use fixed local coordinates agreeing with the principal frame at ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) ∂ t v i \partial_t v_i ∂ t v i
∂ t f = κ 2 ( ∂ t h 11 + 2 G κ 1 2 ) + κ 1 ( ∂ t h 22 + 2 G κ 2 2 ) = κ 2 ∂ t h 1 1 + κ 1 ∂ t h 2 2 . \begin{aligned}
\partial_t f&=\kappa_2(\partial_t h_{11}+2G\kappa_1^2)+\kappa_1(\partial_t h_{22}+2G\kappa_2^2)\\
&=\kappa_2\partial_t h_1^1+\kappa_1\partial_t h_2^2.
\end{aligned} ∂ t f = κ 2 ( ∂ t h 11 + 2 G κ 1 2 ) + κ 1 ( ∂ t h 22 + 2 G κ 2 2 ) = κ 2 ∂ t h 1 1 + κ 1 ∂ t h 2 2 . Contracting (47) with G ˙ m = F ˙ m \dot G^m=\dot F^m G ˙ m = F ˙ m (37) , we obtain
( ∂ t − L ) f = Q [ G ] + R (\partial_t-\mathcal{L})f=Q[G]+\mathcal{R} ( ∂ t − L ) f = Q [ G ] + R at the contact point.
Finally, f − φ ≥ 0 f-\varphi\geq0 f − φ ≥ 0 ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 )
∂ t φ ≥ ∂ t f , ∇ 2 φ ≤ ∇ 2 f . (48) \partial_t\varphi\geq\partial_t f,\qquad\nabla^2\varphi\leq\nabla^2 f. \tag*{(48)} ∂ t φ ≥ ∂ t f , ∇ 2 φ ≤ ∇ 2 f . (48) Since ( G ˙ i j ) (\dot{G}^{ij}) ( G ˙ ij )
( ∂ t − L ) φ ≥ ( ∂ t − L ) f = Q [ G ] + R . (\partial_t-\mathcal{L})\varphi\geq(\partial_t-\mathcal{L})f=Q[G]+\mathcal{R}. ( ∂ t − L ) φ ≥ ( ∂ t − L ) f = Q [ G ] + R . This proves (44) . □
We estimate the derivative terms in (44) for lower supports with zero spatial gradient, as required in the preservation argument. The proof uses the paired concavity inequality in Lemma 6.1, which is proved in § 6 .
Lemma 3.3. Under the assumptions of Lemma 3.2, suppose that ∇ φ ( x 0 , t 0 ) = 0 \nabla\varphi(x_0,t_0)=0 ∇ φ ( x 0 , t 0 ) = 0
Q [ G ] = Q [ F ] ≥ 2 F ( κ 2 ( ∇ 1 F ) 2 + κ 1 ( ∇ 2 F ) 2 ) ≥ 0. (49) Q[G]=Q[F]\geq\frac{2}{F}\left(\kappa_2(\nabla_1F)^2+\kappa_1(\nabla_2F)^2\right)\geq0. \tag*{(49)} Q [ G ] = Q [ F ] ≥ F 2 ( κ 2 ( ∇ 1 F ) 2 + κ 1 ( ∇ 2 F ) 2 ) ≥ 0. (49) Consequently,
( ∂ t − L ) φ ≥ R (50) (\partial_t-\mathcal{L})\varphi\geq\mathcal{R} \tag*{(50)} ( ∂ t − L ) φ ≥ R (50) at the contact point.
Proof. We work at the contact point throughout. Equation (18) makes ∇ m h i j \nabla_m h_{ij} ∇ m h ij (42) gives
κ 2 ∇ m h 11 + κ 1 ∇ m h 22 = 0 , m = 1 , … , n . (51) \kappa_2\nabla_mh_{11}+\kappa_1\nabla_mh_{22}=0,\qquad m=1,\ldots,n. \tag*{(51)} κ 2 ∇ m h 11 + κ 1 ∇ m h 22 = 0 , m = 1 , … , n . (51) Set ψ = − F − 1 \psi=-F^{-1} ψ = − F − 1 (15) . Since G = F − φ ( t ) G=F-\varphi(t) G = F − φ ( t ) φ ( t ) \varphi(t) φ ( t )
Q [ G ] = Q [ F ] = F 2 Q [ ψ ] + 2 F ( κ 2 ( ∇ 1 F ) 2 + κ 1 ( ∇ 2 F ) 2 ) . (52) Q[G]=Q[F]=F^2Q[\psi]+\frac{2}{F}\left(\kappa_2(\nabla_1F)^2+\kappa_1(\nabla_2F)^2\right). \tag*{(52)} Q [ G ] = Q [ F ] = F 2 Q [ ψ ] + F 2 ( κ 2 ( ∇ 1 F ) 2 + κ 1 ( ∇ 2 F ) 2 ) . (52) It suffices to prove that Q [ ψ ] ≥ 0 Q[\psi]\geq0 Q [ ψ ] ≥ 0
First suppose that the principal curvatures are distinct. Applying (8) with ξ i j = ∇ m h i j \xi_{ij}=\nabla_mh_{ij} ξ ij = ∇ m h ij m = 1 , 2 m=1,2 m = 1 , 2 (41) as
Q [ ψ ] = κ 2 ∑ i , j = 1 n ψ ¨ i j ∇ 1 h i i ∇ 1 h j j + κ 1 ∑ i , j = 1 n ψ ¨ i j ∇ 2 h i i ∇ 2 h j j + 2 κ 2 ∑ i < j ψ ˙ i − ψ ˙ j κ i − κ j ( ∇ 1 h i j ) 2 + 2 κ 1 ∑ i < j ψ ˙ i − ψ ˙ j κ i − κ j ( ∇ 2 h i j ) 2 − 2 ∑ m = 1 n ψ ˙ m ∇ m h 11 ∇ m h 22 + 2 ∑ m = 1 n ψ ˙ m ( ∇ m h 12 ) 2 + 2 κ 2 ∑ m = 1 n ∑ p = 3 n ψ ˙ m κ p − κ 1 ( ∇ m h 1 p ) 2 + 2 κ 1 ∑ m = 1 n ∑ p = 3 n ψ ˙ m κ p − κ 2 ( ∇ m h 2 p ) 2 . (53) \begin{aligned}
Q[\psi]={}&\kappa_2\sum_{i,j=1}^{n}\ddot{\psi}^{ij}\nabla_1h_{ii}\nabla_1h_{jj}+\kappa_1\sum_{i,j=1}^{n}\ddot{\psi}^{ij}\nabla_2h_{ii}\nabla_2h_{jj}\\
&+2\kappa_2\sum_{i<j}\frac{\dot{\psi}^{i}-\dot{\psi}^{j}}{\kappa_i-\kappa_j}(\nabla_1h_{ij})^2+2\kappa_1\sum_{i<j}\frac{\dot{\psi}^{i}-\dot{\psi}^{j}}{\kappa_i-\kappa_j}(\nabla_2h_{ij})^2\\
&-2\sum_{m=1}^{n}\dot{\psi}^{m}\nabla_mh_{11}\nabla_mh_{22}+2\sum_{m=1}^{n}\dot{\psi}^{m}(\nabla_mh_{12})^2\\
&+2\kappa_2\sum_{m=1}^{n}\sum_{p=3}^{n}\frac{\dot{\psi}^{m}}{\kappa_p-\kappa_1}(\nabla_mh_{1p})^2\\
&+2\kappa_1\sum_{m=1}^{n}\sum_{p=3}^{n}\frac{\dot{\psi}^{m}}{\kappa_p-\kappa_2}(\nabla_mh_{2p})^2. \tag*{(53)}
\end{aligned} Q [ ψ ] = κ 2 i , j = 1 ∑ n ψ ¨ ij ∇ 1 h ii ∇ 1 h j j + κ 1 i , j = 1 ∑ n ψ ¨ ij ∇ 2 h ii ∇ 2 h j j + 2 κ 2 i < j ∑ κ i − κ j ψ ˙ i − ψ ˙ j ( ∇ 1 h ij ) 2 + 2 κ 1 i < j ∑ κ i − κ j ψ ˙ i − ψ ˙ j ( ∇ 2 h ij ) 2 − 2 m = 1 ∑ n ψ ˙ m ∇ m h 11 ∇ m h 22 + 2 m = 1 ∑ n ψ ˙ m ( ∇ m h 12 ) 2 + 2 κ 2 m = 1 ∑ n p = 3 ∑ n κ p − κ 1 ψ ˙ m ( ∇ m h 1 p ) 2 + 2 κ 1 m = 1 ∑ n p = 3 ∑ n κ p − κ 2 ψ ˙ m ( ∇ m h 2 p ) 2 . (53) Here ψ ¨ i j \ddot{\psi}^{ij} ψ ¨ ij ∇ h \nabla h ∇ h
Q [ ψ ] = I 1 + I 2 + I 3 . (54) Q[\psi]=\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3. \tag*{(54)} Q [ ψ ] = I 1 + I 2 + I 3 . (54) The first group consists of all terms involving ∇ 1 h i i \nabla_1h_{ii} ∇ 1 h ii (53) . By (18) ,
∇ 2 h 12 = ∇ 1 h 22 , ∇ p h 1 p = ∇ 1 h p p , p > 2. \nabla_2h_{12}=\nabla_1h_{22},\qquad\nabla_ph_{1p}=\nabla_1h_{pp},\quad p>2. ∇ 2 h 12 = ∇ 1 h 22 , ∇ p h 1 p = ∇ 1 h pp , p > 2. Thus the ( 1 , 2 ) (1,2) ( 1 , 2 ) m = 2 m=2 m = 2
2 ( κ 1 ψ ˙ 1 − ψ ˙ 2 κ 1 − κ 2 + ψ ˙ 2 ) ( ∇ 1 h 22 ) 2 = 2 κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 κ 1 − κ 2 ( ∇ 1 h 22 ) 2 . 2\left(\kappa_1\frac{\dot{\psi}^{1}-\dot{\psi}^{2}}{\kappa_1-\kappa_2}+\dot{\psi}^{2}\right)(\nabla_1h_{22})^2=2\frac{\kappa_1\dot{\psi}^{1}-\kappa_2\dot{\psi}^{2}}{\kappa_1-\kappa_2}(\nabla_1h_{22})^2. 2 ( κ 1 κ 1 − κ 2 ψ ˙ 1 − ψ ˙ 2 + ψ ˙ 2 ) ( ∇ 1 h 22 ) 2 = 2 κ 1 − κ 2 κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ( ∇ 1 h 22 ) 2 . The product with m = 1 m=1 m = 1 m = p > 2 m=p>2 m = p > 2
I 1 = κ 2 ∑ i , j = 1 n ψ ¨ i j ∇ 1 h i i ∇ 1 h j j + 2 κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 κ 1 − κ 2 ( ∇ 1 h 22 ) 2 − 2 ψ ˙ 1 ∇ 1 h 11 ∇ 1 h 22 + 2 κ 2 ∑ p = 3 n ψ ˙ p κ p − κ 1 ( ∇ 1 h p p ) 2 . (55) \mathcal{I}_1=\kappa_2\sum_{i,j=1}^{n}\ddot{\psi}^{ij}\nabla_1h_{ii}\nabla_1h_{jj}+2\frac{\kappa_1\dot{\psi}^{1}-\kappa_2\dot{\psi}^{2}}{\kappa_1-\kappa_2}(\nabla_1h_{22})^2
-2\dot{\psi}^{1}\nabla_1h_{11}\nabla_1h_{22}+2\kappa_2\sum_{p=3}^{n}\frac{\dot{\psi}^{p}}{\kappa_p-\kappa_1}(\nabla_1h_{pp})^2. \tag*{(55)} I 1 = κ 2 i , j = 1 ∑ n ψ ¨ ij ∇ 1 h ii ∇ 1 h j j + 2 κ 1 − κ 2 κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ( ∇ 1 h 22 ) 2 − 2 ψ ˙ 1 ∇ 1 h 11 ∇ 1 h 22 + 2 κ 2 p = 3 ∑ n κ p − κ 1 ψ ˙ p ( ∇ 1 h pp ) 2 . (55) The second group consists of all terms involving ∇ 2 h i i \nabla_2h_{ii} ∇ 2 h ii ( 1 , 2 ) (1,2) ( 1 , 2 ) m = 1 m=1 m = 1 m = 2 m=2 m = 2 m = p > 2 m=p>2 m = p > 2
∇ 1 h 12 = ∇ 2 h 11 , ∇ p h 2 p = ∇ 2 h p p , p > 2 , \nabla_1h_{12}=\nabla_2h_{11},\qquad\nabla_ph_{2p}=\nabla_2h_{pp},\qquad p>2, ∇ 1 h 12 = ∇ 2 h 11 , ∇ p h 2 p = ∇ 2 h pp , p > 2 , and
κ 2 ψ ˙ 1 − ψ ˙ 2 κ 1 − κ 2 + ψ ˙ 1 = κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 κ 1 − κ 2 , \kappa_2\frac{\dot{\psi}^{1}-\dot{\psi}^{2}}{\kappa_1-\kappa_2}+\dot{\psi}^{1}=\frac{\kappa_1\dot{\psi}^{1}-\kappa_2\dot{\psi}^{2}}{\kappa_1-\kappa_2}, κ 2 κ 1 − κ 2 ψ ˙ 1 − ψ ˙ 2 + ψ ˙ 1 = κ 1 − κ 2 κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 , we obtain
I 2 = κ 1 ∑ i , j = 1 n ψ ¨ i j ∇ 2 h i i ∇ 2 h j j + 2 κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 κ 1 − κ 2 ( ∇ 2 h 11 ) 2 − 2 ψ ˙ 2 ∇ 2 h 11 ∇ 2 h 22 + 2 κ 1 ∑ p = 3 n ψ ˙ p κ p − κ 2 ( ∇ 2 h p p ) 2 . (56) \mathcal{I}_2=\kappa_1\sum_{i,j=1}^{n}\ddot{\psi}^{ij}\nabla_2h_{ii}\nabla_2h_{jj}+2\frac{\kappa_1\dot{\psi}^{1}-\kappa_2\dot{\psi}^{2}}{\kappa_1-\kappa_2}(\nabla_2h_{11})^2
-2\dot{\psi}^{2}\nabla_2h_{11}\nabla_2h_{22}+2\kappa_1\sum_{p=3}^{n}\frac{\dot{\psi}^{p}}{\kappa_p-\kappa_2}(\nabla_2h_{pp})^2. \tag*{(56)} I 2 = κ 1 i , j = 1 ∑ n ψ ¨ ij ∇ 2 h ii ∇ 2 h j j + 2 κ 1 − κ 2 κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ( ∇ 2 h 11 ) 2 − 2 ψ ˙ 2 ∇ 2 h 11 ∇ 2 h 22 + 2 κ 1 p = 3 ∑ n κ p − κ 2 ψ ˙ p ( ∇ 2 h pp ) 2 . (56) The remaining components are
∇ p h 11 , ∇ p h 22 , ∇ p h 12 ( p > 2 ) , ∇ p h 1 r , ∇ p h 2 r ( 3 ≤ p < r ≤ n ) . \nabla_ph_{11},\qquad\nabla_ph_{22},\qquad\nabla_ph_{12}\quad(p>2),\qquad\nabla_ph_{1r},\qquad\nabla_ph_{2r}\quad(3\leq p<r\leq n). ∇ p h 11 , ∇ p h 22 , ∇ p h 12 ( p > 2 ) , ∇ p h 1 r , ∇ p h 2 r ( 3 ≤ p < r ≤ n ) . They form I 3 \mathcal{I}_3 I 3 i = 1 , 2 i=1,2 i = 1 , 2 p > 2 p>2 p > 2 (18) gives ∇ i h i p = ∇ p h i i \nabla_ih_{ip}=\nabla_ph_{ii} ∇ i h i p = ∇ p h ii m = i m=i m = i
ψ ˙ i − ψ ˙ p κ i − κ p + ψ ˙ i κ p − κ i = ψ ˙ p κ p − κ i . \frac{\dot{\psi}^{i}-\dot{\psi}^{p}}{\kappa_i-\kappa_p}+\frac{\dot{\psi}^{i}}{\kappa_p-\kappa_i}=\frac{\dot{\psi}^{p}}{\kappa_p-\kappa_i}. κ i − κ p ψ ˙ i − ψ ˙ p + κ p − κ i ψ ˙ i = κ p − κ i ψ ˙ p . Together with the product having m = p m=p m = p (57) below. For the component ∇ p h 12 = ∇ 1 h 2 p = ∇ 2 h 1 p \nabla_ph_{12}=\nabla_1h_{2p}=\nabla_2h_{1p} ∇ p h 12 = ∇ 1 h 2 p = ∇ 2 h 1 p m = p m=p m = p ( m , i ) = ( 1 , 2 ) (m,i)=(1,2) ( m , i ) = ( 1 , 2 ) ( 2 , 1 ) (2,1) ( 2 , 1 )
κ 2 ψ ˙ 2 − ψ ˙ p κ 2 − κ p + κ 1 ψ ˙ 1 − ψ ˙ p κ 1 − κ p + ψ ˙ p + κ 1 ψ ˙ 1 κ p − κ 2 + κ 2 ψ ˙ 2 κ p − κ 1 = ( κ 2 − κ 1 ) ( κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ) + ( κ p 2 − κ 1 κ 2 ) ψ ˙ p ( κ p − κ 1 ) ( κ p − κ 2 ) . \kappa_2\frac{\dot{\psi}^{2}-\dot{\psi}^{p}}{\kappa_2-\kappa_p}+\kappa_1\frac{\dot{\psi}^{1}-\dot{\psi}^{p}}{\kappa_1-\kappa_p}+\dot{\psi}^{p}+\frac{\kappa_1\dot{\psi}^{1}}{\kappa_p-\kappa_2}+\frac{\kappa_2\dot{\psi}^{2}}{\kappa_p-\kappa_1}
=\frac{(\kappa_2-\kappa_1)(\kappa_1\dot{\psi}^{1}-\kappa_2\dot{\psi}^{2})+(\kappa_p^2-\kappa_1\kappa_2)\dot{\psi}^{p}}{(\kappa_p-\kappa_1)(\kappa_p-\kappa_2)}. κ 2 κ 2 − κ p ψ ˙ 2 − ψ ˙ p + κ 1 κ 1 − κ p ψ ˙ 1 − ψ ˙ p + ψ ˙ p + κ p − κ 2 κ 1 ψ ˙ 1 + κ p − κ 1 κ 2 ψ ˙ 2 = ( κ p − κ 1 ) ( κ p − κ 2 ) ( κ 2 − κ 1 ) ( κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ) + ( κ p 2 − κ 1 κ 2 ) ψ ˙ p . Finally, for i = 1 , 2 i=1,2 i = 1 , 2 3 ≤ p < r ≤ n 3\leq p<r\leq n 3 ≤ p < r ≤ n
∇ i h p r = ∇ p h i r = ∇ r h i p . \nabla_ih_{pr}=\nabla_ph_{ir}=\nabla_rh_{ip}. ∇ i h p r = ∇ p h i r = ∇ r h i p . The ( p , r ) (p,r) ( p , r ) i i i p p p r r r
I 3 = 2 ∑ p = 3 n ψ ˙ p [ κ 2 κ p − κ 1 ( ∇ p h 11 ) 2 + κ 1 κ p − κ 2 ( ∇ p h 22 ) 2 − ∇ p h 11 ∇ p h 22 ] + 2 ∑ p = 3 n ( κ 2 − κ 1 ) ( κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ) + ( κ p 2 − κ 1 κ 2 ) ψ ˙ p ( κ p − κ 1 ) ( κ p − κ 2 ) ( ∇ p h 12 ) 2 + 2 ∑ i = 1 2 ∑ 3 ≤ p < r ≤ n κ 1 κ 2 κ i [ ψ ˙ p − ψ ˙ r κ p − κ r + ψ ˙ p κ r − κ i + ψ ˙ r κ p − κ i ] ( ∇ p h i r ) 2 . (57) \begin{aligned}
\mathcal{I}_3={}&2\sum_{p=3}^{n}\dot{\psi}^{p}\left[\frac{\kappa_2}{\kappa_p-\kappa_1}(\nabla_p h_{11})^2+\frac{\kappa_1}{\kappa_p-\kappa_2}(\nabla_p h_{22})^2-\nabla_p h_{11}\nabla_p h_{22}\right]\\
&+2\sum_{p=3}^{n}\frac{(\kappa_2-\kappa_1)(\kappa_1\dot{\psi}^{1}-\kappa_2\dot{\psi}^{2})+(\kappa_p^2-\kappa_1\kappa_2)\dot{\psi}^{p}}{(\kappa_p-\kappa_1)(\kappa_p-\kappa_2)}(\nabla_p h_{12})^2\\
&+2\sum_{i=1}^{2}\sum_{3\le p<r\le n}\frac{\kappa_1\kappa_2}{\kappa_i}\left[\frac{\dot{\psi}^{p}-\dot{\psi}^{r}}{\kappa_p-\kappa_r}+\frac{\dot{\psi}^{p}}{\kappa_r-\kappa_i}+\frac{\dot{\psi}^{r}}{\kappa_p-\kappa_i}\right](\nabla_p h_{ir})^2. \tag*{(57)}
\end{aligned} I 3 = 2 p = 3 ∑ n ψ ˙ p [ κ p − κ 1 κ 2 ( ∇ p h 11 ) 2 + κ p − κ 2 κ 1 ( ∇ p h 22 ) 2 − ∇ p h 11 ∇ p h 22 ] + 2 p = 3 ∑ n ( κ p − κ 1 ) ( κ p − κ 2 ) ( κ 2 − κ 1 ) ( κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ) + ( κ p 2 − κ 1 κ 2 ) ψ ˙ p ( ∇ p h 12 ) 2 + 2 i = 1 ∑ 2 3 ≤ p < r ≤ n ∑ κ i κ 1 κ 2 [ κ p − κ r ψ ˙ p − ψ ˙ r + κ r − κ i ψ ˙ p + κ p − κ i ψ ˙ r ] ( ∇ p h i r ) 2 . (57) No component with all three indices greater than two occurs in (53) . Thus these three groups account for every term, and we estimate them in this order.
The group I 1 \mathcal{I}_1 I 1 For a vector v ∈ R n v\in\mathbb{R}^n v ∈ R n κ 2 v 1 + κ 1 v 2 = 0 \kappa_2v_1+\kappa_1v_2=0 κ 2 v 1 + κ 1 v 2 = 0
B ( v ) = ∑ i , j ψ ¨ i j v i v j + 2 κ 1 2 ψ ˙ 1 − κ 2 2 ψ ˙ 2 κ 1 − κ 2 ( v 1 κ 1 ) 2 + 2 ∑ p > 2 ψ ˙ p κ p − κ 1 v p 2 . (58) B(v)=\sum_{i,j}\ddot{\psi}^{ij}v_iv_j+2\frac{\kappa_1^2\dot{\psi}^{1}-\kappa_2^2\dot{\psi}^{2}}{\kappa_1-\kappa_2}\left(\frac{v_1}{\kappa_1}\right)^2+2\sum_{p>2}\frac{\dot{\psi}^{p}}{\kappa_p-\kappa_1}v_p^2. \tag*{(58)} B ( v ) = i , j ∑ ψ ¨ ij v i v j + 2 κ 1 − κ 2 κ 1 2 ψ ˙ 1 − κ 2 2 ψ ˙ 2 ( κ 1 v 1 ) 2 + 2 p > 2 ∑ κ p − κ 1 ψ ˙ p v p 2 . (58) By (55) and (51) ,
I 1 = κ 2 B ( ∇ 1 h 11 , … , ∇ 1 h n n ) . (59) \mathcal{I}_1=\kappa_2B(\nabla_1h_{11},\ldots,\nabla_1h_{nn}). \tag*{(59)} I 1 = κ 2 B ( ∇ 1 h 11 , … , ∇ 1 h nn ) . (59) To show that B ( v ) ≥ 0 B(v)\ge0 B ( v ) ≥ 0
r = n − ℓ , s = n − k , λ i = κ i − 1 , ξ i = − v i κ i 2 . r=n-\ell,\qquad s=n-k,\qquad\lambda_i=\kappa_i^{-1},\qquad\xi_i=-\frac{v_i}{\kappa_i^2}. r = n − ℓ , s = n − k , λ i = κ i − 1 , ξ i = − κ i 2 v i . Then λ 1 > λ 2 ≥ ⋯ ≥ λ n > 0 \lambda_1>\lambda_2\ge\cdots\ge\lambda_n>0 λ 1 > λ 2 ≥ ⋯ ≥ λ n > 0 λ 2 ξ 1 + λ 1 ξ 2 = 0 \lambda_2\xi_1+\lambda_1\xi_2=0 λ 2 ξ 1 + λ 1 ξ 2 = 0 q r , s = ( σ r σ s ) 1 / ( r − s ) q_{r,s}=\left(\frac{\sigma_r}{\sigma_s}\right)^{1/(r-s)} q r , s = ( σ s σ r ) 1/ ( r − s ) Proposition 6.1 , (11) gives
F ∗ ( λ ) = ( n s ) 1 / ( r − s ) q r , s ( λ ) , ψ ( κ ) = − F ∗ ( λ ) . F_*(\lambda)=\binom{n}{s}^{1/(r-s)}q_{r,s}(\lambda),\qquad\psi(\kappa)=-F_*(\lambda). F ∗ ( λ ) = ( s n ) 1/ ( r − s ) q r , s ( λ ) , ψ ( κ ) = − F ∗ ( λ ) . Consequently,
ψ ˙ i = ∂ i F ∗ κ i 2 , ψ ¨ i j = − ∂ i j F ∗ κ i 2 κ j 2 − 2 ∂ i F ∗ κ i 3 δ i j . \dot{\psi}^{i}=\frac{\partial_iF_*}{\kappa_i^2},\qquad\ddot{\psi}^{ij}=-\frac{\partial_{ij}F_*}{\kappa_i^2\kappa_j^2}-\frac{2\partial_iF_*}{\kappa_i^3}\delta_{ij}. ψ ˙ i = κ i 2 ∂ i F ∗ , ψ ¨ ij = − κ i 2 κ j 2 ∂ ij F ∗ − κ i 3 2 ∂ i F ∗ δ ij . Substitution into (58) , using the relation between ξ 1 \xi_1 ξ 1 ξ 2 \xi_2 ξ 2
B ( v ) = − ∂ ξ 2 F ∗ + 2 ∑ i = 2 n ∂ i F ∗ λ 1 − λ i ξ i 2 − 2 ∂ 1 F ∗ λ 1 − λ 2 ξ 1 2 ≥ 0. (60) B(v)=-\partial_{\xi}^{2}F_*+2\sum_{i=2}^{n}\frac{\partial_iF_*}{\lambda_1-\lambda_i}\xi_i^2-\frac{2\partial_1F_*}{\lambda_1-\lambda_2}\xi_1^2\ge0. \tag*{(60)} B ( v ) = − ∂ ξ 2 F ∗ + 2 i = 2 ∑ n λ 1 − λ i ∂ i F ∗ ξ i 2 − λ 1 − λ 2 2 ∂ 1 F ∗ ξ 1 2 ≥ 0. (60) The last inequality follows from (99) in Proposition 6.1 , since 1 ≤ s < r ≤ n − 1 1\le s<r\le n-1 1 ≤ s < r ≤ n − 1 r + s = 2 n − k − ℓ ≥ n r+s=2n-k-\ell\ge n r + s = 2 n − k − ℓ ≥ n I 1 ≥ 0 \mathcal{I}_1\ge0 I 1 ≥ 0
The group I 2 \mathcal{I}_2 I 2 By (51) , ∇ 2 h 22 / κ 2 = − ∇ 2 h 11 / κ 1 \nabla_2h_{22}/\kappa_2=-\nabla_2h_{11}/\kappa_1 ∇ 2 h 22 / κ 2 = − ∇ 2 h 11 / κ 1 (56) gives
I 2 κ 1 = B ( ∇ 2 h 11 , … , ∇ 2 h n n ) + 2 ∑ p > 2 ψ ˙ p ( 1 κ p − κ 2 − 1 κ p − κ 1 ) ( ∇ 2 h p p ) 2 ≥ 0. (61) \frac{\mathcal{I}_2}{\kappa_1}=B(\nabla_2h_{11},\ldots,\nabla_2h_{nn})+2\sum_{p>2}\dot{\psi}^{p}\left(\frac{1}{\kappa_p-\kappa_2}-\frac{1}{\kappa_p-\kappa_1}\right)(\nabla_2h_{pp})^2\ge0. \tag*{(61)} κ 1 I 2 = B ( ∇ 2 h 11 , … , ∇ 2 h nn ) + 2 p > 2 ∑ ψ ˙ p ( κ p − κ 2 1 − κ p − κ 1 1 ) ( ∇ 2 h pp ) 2 ≥ 0. (61) The first term is nonnegative by (60) , and every coefficient in the remaining sum is nonnegative.
The group I 3 \mathcal{I}_3 I 3 The first sum in (57) is nonnegative, since (51) gives ( ∇ p h 11 ) ( ∇ p h 22 ) ≤ 0 (\nabla_ph_{11})(\nabla_ph_{22})\le0 ( ∇ p h 11 ) ( ∇ p h 22 ) ≤ 0 p > 2 p>2 p > 2 (16) implies
ψ ˙ 1 ≥ ψ ˙ 2 ≥ ψ ˙ p > 0 , κ p 2 ψ ˙ p ≥ κ 2 2 ψ ˙ 2 . \dot{\psi}^{1}\ge\dot{\psi}^{2}\ge\dot{\psi}^{p}>0,\qquad\kappa_p^2\dot{\psi}^{p}\ge\kappa_2^2\dot{\psi}^{2}. ψ ˙ 1 ≥ ψ ˙ 2 ≥ ψ ˙ p > 0 , κ p 2 ψ ˙ p ≥ κ 2 2 ψ ˙ 2 . Hence its numerator satisfies
( κ 2 − κ 1 ) ( κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ) + ( κ p 2 − κ 1 κ 2 ) ψ ˙ p ≥ − ( κ 2 − κ 1 ) 2 ψ ˙ 2 + ( κ 2 2 − κ 1 κ 2 ) ψ ˙ 2 = κ 1 ( κ 2 − κ 1 ) ψ ˙ 2 ≥ 0. (62) (\kappa_2-\kappa_1)(\kappa_1\dot\psi^1-\kappa_2\dot\psi^2)+(\kappa_p^2-\kappa_1\kappa_2)\dot\psi^p
\geq-(\kappa_2-\kappa_1)^2\dot\psi^2+(\kappa_2^2-\kappa_1\kappa_2)\dot\psi^2
=\kappa_1(\kappa_2-\kappa_1)\dot\psi^2\geq0. \tag*{(62)} ( κ 2 − κ 1 ) ( κ 1 ψ ˙ 1 − κ 2 ψ ˙ 2 ) + ( κ p 2 − κ 1 κ 2 ) ψ ˙ p ≥ − ( κ 2 − κ 1 ) 2 ψ ˙ 2 + ( κ 2 2 − κ 1 κ 2 ) ψ ˙ 2 = κ 1 ( κ 2 − κ 1 ) ψ ˙ 2 ≥ 0. (62) For the last sum, the same identities give
ψ ˙ p − ψ ˙ r κ p − κ r + ψ ˙ p κ r − κ i + ψ ˙ r κ p − κ i ≥ ψ ˙ p − ψ ˙ r κ p − κ r + ψ ˙ p κ r + ψ ˙ r κ p ≥ 0 , i = 1 , 2 , 3 ≤ p < r ≤ n . \frac{\dot\psi^p-\dot\psi^r}{\kappa_p-\kappa_r}+\frac{\dot\psi^p}{\kappa_r-\kappa_i}+\frac{\dot\psi^r}{\kappa_p-\kappa_i}\geq\frac{\dot\psi^p-\dot\psi^r}{\kappa_p-\kappa_r}+\frac{\dot\psi^p}{\kappa_r}+\frac{\dot\psi^r}{\kappa_p}\geq0,\qquad i=1,2,\quad3\leq p<r\leq n. κ p − κ r ψ ˙ p − ψ ˙ r + κ r − κ i ψ ˙ p + κ p − κ i ψ ˙ r ≥ κ p − κ r ψ ˙ p − ψ ˙ r + κ r ψ ˙ p + κ p ψ ˙ r ≥ 0 , i = 1 , 2 , 3 ≤ p < r ≤ n . Thus I 3 ≥ 0 I_3\geq0 I 3 ≥ 0
It remains to consider repeated principal curvatures. Set T i j k = ∇ i h j k T_{ijk}=\nabla_i h_{jk} T ij k = ∇ i h j k ε > 0 \varepsilon>0 ε > 0
κ i ε = κ i + i ε , T i j k ε = ( κ i ε κ j ε κ k ε κ i κ j κ k ) 1 / 2 T i j k , \kappa_i^\varepsilon=\kappa_i+i\varepsilon,\qquad T_{ijk}^\varepsilon=\left(\frac{\kappa_i^\varepsilon\kappa_j^\varepsilon\kappa_k^\varepsilon}{\kappa_i\kappa_j\kappa_k}\right)^{1/2}T_{ijk}, κ i ε = κ i + i ε , T ij k ε = ( κ i κ j κ k κ i ε κ j ε κ k ε ) 1/2 T ij k , with no summation in this definition. By (18) , T ε T^\varepsilon T ε ( κ ε , T ε ) → ( κ , T ) (\kappa^\varepsilon,T^\varepsilon)\to(\kappa,T) ( κ ε , T ε ) → ( κ , T ) (51) gives
κ 2 ε T m 11 ε + κ 1 ε T m 22 ε = ( κ m ε κ m ) 1 / 2 κ 1 ε κ 2 ε κ 1 κ 2 ( κ 2 T m 11 + κ 1 T m 22 ) = 0. \kappa_2^\varepsilon T_{m11}^\varepsilon+\kappa_1^\varepsilon T_{m22}^\varepsilon=\left(\frac{\kappa_m^\varepsilon}{\kappa_m}\right)^{1/2}\frac{\kappa_1^\varepsilon\kappa_2^\varepsilon}{\kappa_1\kappa_2}(\kappa_2T_{m11}+\kappa_1T_{m22})=0. κ 2 ε T m 11 ε + κ 1 ε T m 22 ε = ( κ m κ m ε ) 1/2 κ 1 κ 2 κ 1 ε κ 2 ε ( κ 2 T m 11 + κ 1 T m 22 ) = 0. Thus the preceding calculation applies to these data and gives Q ε [ ψ ] ≥ 0 Q^\varepsilon[\psi]\geq0 Q ε [ ψ ] ≥ 0 Q ε Q^\varepsilon Q ε (41) evaluated at ( κ ε , T ε ) (\kappa^\varepsilon,T^\varepsilon) ( κ ε , T ε )
For i = 1 , 2 i=1,2 i = 1 , 2 p > 2 p>2 p > 2 m m m (43) implies
( T m i p ε ) 2 κ p ε − κ i ε ⟶ { T m i p 2 κ p − κ i , p ∈ J i , 0 , κ p = κ i . \frac{(T_{mip}^\varepsilon)^2}{\kappa_p^\varepsilon-\kappa_i^\varepsilon}\longrightarrow
\begin{cases}
\frac{T_{mip}^2}{\kappa_p-\kappa_i}, & p\in J_i,\\
0, & \kappa_p=\kappa_i.
\end{cases} κ p ε − κ i ε ( T mi p ε ) 2 ⟶ { κ p − κ i T mi p 2 , 0 , p ∈ J i , κ p = κ i . In the second case the numerator vanishes for every ε \varepsilon ε ψ \psi ψ
Q [ ψ ] = lim ε → 0 + Q ε [ ψ ] ≥ 0. Q[\psi]=\lim_{\varepsilon\to0^+}Q^\varepsilon[\psi]\geq0. Q [ ψ ] = ε → 0 + lim Q ε [ ψ ] ≥ 0. Combining this with (52) proves (49) . Equation (44) then gives (50) . □
Proof of Theorem 3.1 . Euler’s identity F = ∑ i F ˙ i κ i F=\sum_i\dot F^i\kappa_i F = ∑ i F ˙ i κ i (45) as
R = 2 ∑ i F ˙ i ( κ i − κ 1 ) ( κ i − κ 2 ) + S [ 2 F ˙ i j ( h 2 ) i j − ϕ ( κ 1 + κ 2 ) ] . (63) \mathcal{R}=2\sum_i\dot F^i(\kappa_i-\kappa_1)(\kappa_i-\kappa_2)+S\left[2\dot F^{ij}(h^2)_{ij}-\phi(\kappa_1+\kappa_2)\right]. \tag*{(63)} R = 2 i ∑ F ˙ i ( κ i − κ 1 ) ( κ i − κ 2 ) + S [ 2 F ˙ ij ( h 2 ) ij − ϕ ( κ 1 + κ 2 ) ] . (63) The first sum is nonnegative. Fix τ < T \tau<T τ < T
C τ = max M × [ 0 , τ ] ϕ ( κ 1 + κ 2 ) . C_\tau=\max_{M\times[0,\tau]}\phi(\kappa_1+\kappa_2). C τ = M × [ 0 , τ ] max ϕ ( κ 1 + κ 2 ) . Since F ˙ i j ( h 2 ) i j > 0 \dot F^{ij}(h^2)_{ij}>0 F ˙ ij ( h 2 ) ij > 0
R ≥ − C τ S whenever S ≥ 0. (64) \mathcal{R}\geq-C_\tau S\qquad\text{whenever }S\geq0. \tag*{(64)} R ≥ − C τ S whenever S ≥ 0. (64) Choose 0 < ε 0 < min M S ( ⋅ , 0 ) 0<\varepsilon_0<\min_M S(\cdot,0) 0 < ε 0 < min M S ( ⋅ , 0 )
b ( t ) = ε 0 e − ( C τ + 1 ) t . b(t)=\varepsilon_0e^{-(C_\tau+1)t}. b ( t ) = ε 0 e − ( C τ + 1 ) t . Suppose that S S S b b b ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) 0 < t 0 ≤ τ 0<t_0\leq\tau 0 < t 0 ≤ τ ϕ ( x , t ) = b ( t ) \phi(x,t)=b(t) ϕ ( x , t ) = b ( t ) S S S ∇ ϕ = 0 \nabla\phi=0 ∇ ϕ = 0 (64) , give
− ( C τ + 1 ) b ( t 0 ) = ( ∂ t − L ) ϕ ( x 0 , t 0 ) ≥ R ( x 0 , t 0 ) ≥ − C τ b ( t 0 ) , -(C_\tau+1)b(t_0)=(\partial_t-\mathcal{L})\phi(x_0,t_0)\geq\mathcal{R}(x_0,t_0)\geq-C_\tau b(t_0), − ( C τ + 1 ) b ( t 0 ) = ( ∂ t − L ) ϕ ( x 0 , t 0 ) ≥ R ( x 0 , t 0 ) ≥ − C τ b ( t 0 ) , a contradiction. Thus S ≥ b ( t ) > 0 S\geq b(t)>0 S ≥ b ( t ) > 0 M × [ 0 , τ ] M\times[0,\tau] M × [ 0 , τ ] τ < T \tau<T τ < T
Remark 3.4. For 1 ≤ ℓ < k 1 \le\ell< k 1 ≤ ℓ < k k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n F F F (7) . Consider the flow
∂ t X = ( a ( t ) F − F ) ν , a ( t ) = ∫ M t E ℓ F d μ t ∫ M t E ℓ / F d μ t . \partial_t X = \left(\frac{a(t)}{F}-F\right)\nu,\qquad a(t)=\frac{\int_{M_t} E_\ell F\,\mathrm{d}\mu_t}{\int_{M_t} E_\ell/F\,\mathrm{d}\mu_t}. ∂ t X = ( F a ( t ) − F ) ν , a ( t ) = ∫ M t E ℓ / F d μ t ∫ M t E ℓ F d μ t . For k = ℓ + 1 k=\ell+1 k = ℓ + 1 [26 ] . The choice of a ( t ) a(t) a ( t ) W ℓ W_\ell W ℓ W k W_k W k (28) . Positive sectional curvature is also preserved along smooth strictly convex solutions. Indeed, at a lower-support contact with ∇ φ = 0 \nabla\varphi=0 ∇ φ = 0 S ≥ 0 S\ge0 S ≥ 0 Lemma 3.3 gives Q [ F − a / F ] = Q [ F ] + a Q [ − F − 1 ] ≥ 0 Q[F-a/F]=Q[F]+aQ[-F^{-1}]\ge0 Q [ F − a / F ] = Q [ F ] + a Q [ − F − 1 ] ≥ 0 L ^ = ( 1 + a / F 2 ) F ˙ i j ∇ i ∇ j \widehat{\mathcal{L}}=(1+a/F^2)\dot{F}^{ij}\nabla_i\nabla_j L = ( 1 + a / F 2 ) F ˙ ij ∇ i ∇ j
( ∂ t − L ^ ) φ ≥ − 2 a F ( κ 1 + κ 2 ) S . (\partial_t-\widehat{\mathcal{L}})\varphi\ge-\frac{2a}{F}(\kappa_1+\kappa_2)S. ( ∂ t − L ) φ ≥ − F 2 a ( κ 1 + κ 2 ) S . The coefficient on the right is bounded on each compact smooth time interval, so the barrier argument in the proof of Theorem 3.1 applies.
Long-time existence and convergence
Let [ 0 , T max ) [0,T_{\max}) [ 0 , T m a x ) M 0 M_0 M 0 Theorem 1.3 . Theorem 3.1 gives positive sectional curvature on this interval. We use (30) to obtain the geometric estimates, long-time existence, and convergence.
Shape estimate
For a bounded convex domain Ω \Omega Ω ρ − ( Ω ) \rho_-(\Omega) ρ − ( Ω ) ρ + ( Ω ) \rho_+(\Omega) ρ + ( Ω ) [2 Section 4.1] . The preserved quermassintegral replaces the enclosed volume in the containment argument.
Lemma 4.1. There are a fixed point p ∈ H n + 1 p \in\mathbb{H}^{n+1} p ∈ H n + 1 R 0 R_0 R 0 ℓ \ell ℓ
Ω t ⊂ B R 0 ( p ) \Omega_t \subset B_{R_0}(p) Ω t ⊂ B R 0 ( p ) for every t ∈ [ 0 , T max ) t \in[0,T_{\max}) t ∈ [ 0 , T m a x )
Proof. We use the Alexandrov reflection argument in the proof of [2 Lemma 4.1] . Choose R > 0 R>0 R > 0
Ω ‾ 0 ⊂ B R ( p ) , W ℓ ( Ω 0 ) < W ℓ ( B R ( p ) ) . \overline{\Omega}_0 \subset B_R(p),\qquad W_\ell(\Omega_0)<W_\ell(B_R(p)). Ω 0 ⊂ B R ( p ) , W ℓ ( Ω 0 ) < W ℓ ( B R ( p )) . Since the global term ϕ ( t ) \phi(t) ϕ ( t ) [2 Lemma 4.1] applies without change.
It follows first that Ω t \Omega_t Ω t B R ( p ) B_R(p) B R ( p ) t < T max t<T_{\max} t < T m a x M t M_t M t B R ( p ) B_R(p) B R ( p ) B R ( p ) ⊂ Ω t B_R(p)\subset\Omega_t B R ( p ) ⊂ Ω t W ℓ W_\ell W ℓ
W ℓ ( B R ( p ) ) ≤ W ℓ ( Ω t ) = W ℓ ( Ω 0 ) , W_\ell(B_R(p))\le W_\ell(\Omega_t)=W_\ell(\Omega_0), W ℓ ( B R ( p )) ≤ W ℓ ( Ω t ) = W ℓ ( Ω 0 ) , a contradiction. The remaining reflection argument in [2 Lemma 4.1] gives a uniform diameter bound, and hence
Ω t ⊂ B R 0 ( p ) \Omega_t\subset B_{R_0}(p) Ω t ⊂ B R 0 ( p ) for a constant R 0 R_0 R 0 Ω 0 \Omega_0 Ω 0 ℓ \ell ℓ
The following noncollapsing estimate extends the corresponding argument in [2 Lemma 4.1] , where preservation of the enclosed volume is used to obtain an inradius lower bound. Here we replace volume preservation by a positive lower bound for a general quermassintegral W j W_j W j
Lemma 4.2. Let n ≥ 2 n \ge2 n ≥ 2 D , A > 0 D,A > 0 D , A > 0 j ∈ { 0 , … , n − 1 } j \in\{0,\ldots,n-1\} j ∈ { 0 , … , n − 1 }
r 0 = r 0 ( n , D , A , j ) > 0 r_0=r_0(n,D,A,j)>0 r 0 = r 0 ( n , D , A , j ) > 0 with the following property. Suppose that Ω ⊂ H n + 1 \Omega\subset\mathbb{H}^{n+1} Ω ⊂ H n + 1 M = ∂ Ω M=\partial\Omega M = ∂ Ω κ 1 , … , κ n \kappa_1,\ldots,\kappa_n κ 1 , … , κ n M M M
diam H n + 1 Ω ≤ D , W j ( Ω ) ≥ A , \operatorname{diam}_{\mathbb{H}^{n+1}}\Omega\le D,\qquad W_j(\Omega)\ge A, diam H n + 1 Ω ≤ D , W j ( Ω ) ≥ A , and that M M M
κ i κ ℓ ≥ 1 on M for all i ≠ ℓ . \kappa_i\kappa_\ell\ge1\qquad\text{on }M\qquad\text{for all }i\ne\ell. κ i κ ℓ ≥ 1 on M for all i = ℓ . Then the inner radius
ρ − ( Ω ) = sup { r > 0 ∣ B r ( p ) ⊂ Ω for some p ∈ Ω } \rho_-(\Omega)=\sup\{r>0\mid B_r(p)\subset\Omega\text{ for some }p\in\Omega\} ρ − ( Ω ) = sup { r > 0 ∣ B r ( p ) ⊂ Ω for some p ∈ Ω } satisfies
ρ − ( Ω ) > r 0 . \rho_-(\Omega)>r_0. ρ − ( Ω ) > r 0 . In particular, Ω \Omega Ω r 0 r_0 r 0
Proof. Fix p ∈ Ω p\in\Omega p ∈ Ω (21) centred at p p p K ⊂ B 1 ( 0 ) K\subset B_1(0) K ⊂ B 1 ( 0 ) Ω \Omega Ω
∣ Y ∣ ≤ ρ = tanh D < 1 on K . |Y|\le\rho=\tanh D<1\qquad\text{on }K. ∣ Y ∣ ≤ ρ = tanh D < 1 on K . Since s 2 ≤ ∣ Y ∣ 2 s^2\le|Y|^2 s 2 ≤ ∣ Y ∣ 2 (22) and (23) imply
C − 1 g E ≤ g H ≤ C g E , C − 1 h E ≤ h H ≤ C h E , C^{-1}g_E\le g_H\le Cg_E,\qquad C^{-1}h_E\le h_H\le Ch_E, C − 1 g E ≤ g H ≤ C g E , C − 1 h E ≤ h H ≤ C h E , where C C C n n n D D D 0 < κ ˉ 1 ≤ ⋯ ≤ κ ˉ n 0<\bar\kappa_1\le\cdots\le\bar\kappa_n 0 < κ ˉ 1 ≤ ⋯ ≤ κ ˉ n ∂ K \partial K ∂ K κ 1 κ 2 ≥ 1 \kappa_1\kappa_2\ge1 κ 1 κ 2 ≥ 1
κ ˉ 1 κ ˉ 2 ≥ c ( D ) > 0. (65) \bar\kappa_1\bar\kappa_2\ge c(D)>0. \tag*{(65)} κ ˉ 1 κ ˉ 2 ≥ c ( D ) > 0. (65) Let W j E W_j^E W j E j ≥ 1 j\ge1 j ≥ 1 (27) , the positivity of the lower order quermassintegrals, and (22) –(23) give
W j ( Ω ) ≤ 1 n + 1 − j ∫ ∂ Ω E j − 1 d μ H ≤ C ( n , D ) W j E ( K ) . (66) W_j(\Omega)\le\frac{1}{n+1-j}\int_{\partial\Omega}E_{j-1}\,d\mu_H\le C(n,D)W_j^E(K). \tag*{(66)} W j ( Ω ) ≤ n + 1 − j 1 ∫ ∂ Ω E j − 1 d μ H ≤ C ( n , D ) W j E ( K ) . (66) For j = 0 j=0 j = 0
By John’s ellipsoid theorem [13 ] , there is an ellipsoid E E E K K K
E ⊂ K ⊂ ( n + 1 ) E . E\subset K\subset(n+1)E. E ⊂ K ⊂ ( n + 1 ) E . Let 0 < a 1 ≤ ⋯ ≤ a n + 1 0<a_1\le\cdots\le a_{n+1} 0 < a 1 ≤ ⋯ ≤ a n + 1 ε = a 1 \varepsilon=a_1 ε = a 1 b = a n b=a_n b = a n K K K E E E h E h_E h E ν E \nu_E ν E (65) implies that its Gauss curvature is bounded below by a positive constant depending only on D D D
In the plane spanned by the two longest axes, the projection of the section contains a disk D b D_b D b b b b
z = f ( x ) , x ∈ D b , z=f(x),\qquad x\in D_b, z = f ( x ) , x ∈ D b , with
0 ≤ f ≤ ( n + 1 ) ε . 0\le f\le(n+1)\varepsilon. 0 ≤ f ≤ ( n + 1 ) ε . Since the Gauss curvature of the graph is bounded below by c ( D ) > 0 c(D)>0 c ( D ) > 0
det ( − D 2 f ) ( 1 + ∣ D f ∣ 2 ) 2 ≥ c ( D ) . \frac{\det(-D^2f)}{(1+|Df|^2)^2}\ge c(D). ( 1 + ∣ D f ∣ 2 ) 2 det ( − D 2 f ) ≥ c ( D ) . By the arithmetic–geometric mean inequality,
− Δ f ≥ 2 det ( − D 2 f ) ≥ 2 c ( D ) . -\Delta f \ge2\sqrt{\det(-D^2f)} \ge2\sqrt{c(D)}. − Δ f ≥ 2 det ( − D 2 f ) ≥ 2 c ( D ) . For 0 < R < b 0<R<b 0 < R < b
q R ( x ) = c ( D ) 2 ( R 2 − ∣ x ∣ 2 ) . q_R(x)=\frac{\sqrt{c(D)}}{2}(R^2-|x|^2). q R ( x ) = 2 c ( D ) ( R 2 − ∣ x ∣ 2 ) . Then
− Δ q R = 2 c ( D ) , q R = 0 on ∂ D R . -\Delta q_R=2\sqrt{c(D)},\qquad q_R=0\quad\text{on }\partial D_R. − Δ q R = 2 c ( D ) , q R = 0 on ∂ D R . Since f ≥ 0 f\ge0 f ≥ 0 ∂ D R \partial D_R ∂ D R f ≥ q R f\ge q_R f ≥ q R D R D_R D R
( n + 1 ) ε ≥ f ( 0 ) ≥ c ( D ) 2 R 2 . (n+1)\varepsilon\ge f(0)\ge\frac{\sqrt{c(D)}}{2}R^2. ( n + 1 ) ε ≥ f ( 0 ) ≥ 2 c ( D ) R 2 . Letting R ↑ b R\uparrow b R ↑ b
b 2 ≤ C ( n , D ) ε . (67) b^2\le C(n,D)\varepsilon. \tag*{(67)} b 2 ≤ C ( n , D ) ε . (67) It follows from John’s inclusion that K K K C ε C\sqrt{\varepsilon} C ε I I I β m \beta_m β m R m \mathbb{R}^m R m [31 Section 4.2] ) gives
W j E ( I + B s ) = β n + 1 s n + 1 − j + ∣ I ∣ β n n + 1 − j n + 1 s n − j . W_j^E(I+B_s)=\beta_{n+1}s^{n+1-j}+|I|\beta_n\frac{n+1-j}{n+1}s^{n-j}. W j E ( I + B s ) = β n + 1 s n + 1 − j + ∣ I ∣ β n n + 1 n + 1 − j s n − j . Using Euclidean inclusion monotonicity, (66) , and (67) , we obtain
A ≤ W j ( Ω ) ≤ C ( ε ( n + 1 − j ) / 2 + ε ( n − j ) / 2 ) . A\le W_j(\Omega)\le C\left(\varepsilon^{(n+1-j)/2}+\varepsilon^{(n-j)/2}\right). A ≤ W j ( Ω ) ≤ C ( ε ( n + 1 − j ) /2 + ε ( n − j ) /2 ) . Since 0 < ε ≤ ρ < 1 0<\varepsilon\le\rho<1 0 < ε ≤ ρ < 1 j ≤ n − 1 j\le n-1 j ≤ n − 1
A ≤ C ε ( n − j ) / 2 , ε ≥ ( A C ) 2 / ( n − j ) > 0. A\le C\varepsilon^{(n-j)/2},\qquad\varepsilon\ge\left(\frac{A}{C}\right)^{2/(n-j)}>0. A ≤ C ε ( n − j ) /2 , ε ≥ ( C A ) 2/ ( n − j ) > 0. If z E z_E z E B E ( z E ) ⊂ K B_E(z_E)\subset K B E ( z E ) ⊂ K B ρ ( 0 ) B_\rho(0) B ρ ( 0 ) K K K c ( D ) ε c(D)\varepsilon c ( D ) ε
ρ − ( Ω ) ≥ r 0 ( n , D , A , j ) > 0 \rho_-(\Omega)\ge r_0(n,D,A,j)>0 ρ − ( Ω ) ≥ r 0 ( n , D , A , j ) > 0 and proves the lemma. □
Corollary 4.4. Under the assumptions of Theorem 1.3 , there exist constants r 0 , R 0 > 0 r_0,R_0>0 r 0 , R 0 > 0 n n n k k k ℓ \ell ℓ M 0 M_0 M 0
0 < r 0 ≤ ρ − ( Ω t ) ≤ ρ + ( Ω t ) ≤ R 0 , Ω t ⊂ B R 0 ( p ) , (68) 0<r_0\le\rho_-(\Omega_t)\le\rho_+(\Omega_t)\le R_0,\qquad\Omega_t\subset B_{R_0}(p), \tag*{(68)} 0 < r 0 ≤ ρ − ( Ω t ) ≤ ρ + ( Ω t ) ≤ R 0 , Ω t ⊂ B R 0 ( p ) , (68) for every t ∈ [ 0 , T max ) t\in[0,T_{\max}) t ∈ [ 0 , T m a x ) p p p Lemma 4.1 .
Proof. The conclusion follows from Theorem 3.1 , Lemmas 4.1 and 4.2 , and the preservation of W ℓ W_{\ell} W ℓ 1 ≤ ℓ < k ≤ n − 1 1 \le\ell< k \le n - 1 1 ≤ ℓ < k ≤ n − 1 ℓ ≤ n − 2 \ell\le n - 2 ℓ ≤ n − 2 Lemma 4.2 applies with j = ℓ j = \ell j = ℓ
Lemma 4.5. There is τ 0 > 0 \tau_{0} > 0 τ 0 > 0 Theorem 1.3 , with the following property. Given t 0 < T max t_{0} < T_{\max} t 0 < T m a x B r 0 ( p 0 ) ⊂ Ω t 0 B_{r_{0}}(p_{0}) \subset\Omega_{t_{0}} B r 0 ( p 0 ) ⊂ Ω t 0
B r 0 / 4 ( p 0 ) ⊂ Ω t for t 0 ≤ t < min { T max , t 0 + τ 0 } . B_{r_{0}/4}(p_{0}) \subset\Omega_{t} \quad\text{for } t_{0} \le t < \min\{T_{\max},t_{0}+\tau_{0}\}. B r 0 /4 ( p 0 ) ⊂ Ω t for t 0 ≤ t < min { T m a x , t 0 + τ 0 } . On this time interval, the support function relative to p 0 p_{0} p 0
sinh ( r 0 / 4 ) ≤ u ≤ sinh ( 2 R 0 ) , r ≤ 2 R 0 . (69) \sinh(r_{0}/4) \le u \le\sinh(2R_{0}), \qquad r \le2R_{0}. \tag*{(69)} sinh ( r 0 /4 ) ≤ u ≤ sinh ( 2 R 0 ) , r ≤ 2 R 0 . (69) Proof. Consider the shrinking geodesic ball with radius ρ \rho ρ
ρ ′ = − coth ρ , ρ ( t 0 ) = r 0 / 2. \rho'=-\coth\rho,\qquad\rho(t_{0})=r_{0}/2. ρ ′ = − coth ρ , ρ ( t 0 ) = r 0 /2. It satisfies cosh ρ ( t ) = e − ( t − t 0 ) cosh ( r 0 / 2 ) \cosh\rho(t)=e^{-(t-t_{0})}\cosh(r_{0}/2) cosh ρ ( t ) = e − ( t − t 0 ) cosh ( r 0 /2 )
τ 0 = log cosh ( r 0 / 2 ) cosh ( r 0 / 4 ) . \tau_{0}=\log\frac{\cosh(r_{0}/2)}{\cosh(r_{0}/4)}. τ 0 = log cosh ( r 0 /4 ) cosh ( r 0 /2 ) . At a first contact with the enclosing hypersurface, its Weingarten matrix is at most coth ρ I \coth\rho\, I coth ρ I F F F
ϕ − F > − F ≥ − coth ρ = ρ ′ , \phi-F>-F\ge-\coth\rho=\rho', ϕ − F > − F ≥ − coth ρ = ρ ′ , contradicting a first crossing. Thus the ball remains inside. The support lower bound follows from (19) , and the upper bounds follow from the fixed outer ball and p 0 ∈ Ω t 0 p_{0}\in\Omega_{t_{0}} p 0 ∈ Ω t 0
Upper bound of F F F
We first record consequences of positive sectional curvature. If a = κ 1 < 1 a=\kappa_{1}<1 a = κ 1 < 1 κ i ≥ a − 1 \kappa_{i}\ge a^{-1} κ i ≥ a − 1 i ≥ 2 i\ge2 i ≥ 2 ℓ ≤ j ≤ k − 1 ≤ n − 2 \ell\le j\le k-1\le n-2 ℓ ≤ j ≤ k − 1 ≤ n − 2
E j + 1 E j ≥ n − j − 1 + ( j + 1 ) a 2 a ( n − j + j a 2 ) = 1 + ( 1 − a ) ( j a 2 − a + n − j − 1 ) a ( n − j + j a 2 ) ≥ 1. \frac{E_{j+1}}{E_{j}}\ge\frac{n-j-1+(j+1)a^{2}}{a(n-j+ja^{2})}=1+\frac{(1-a)(ja^{2}-a+n-j-1)}{a(n-j+ja^{2})}\ge1. E j E j + 1 ≥ a ( n − j + j a 2 ) n − j − 1 + ( j + 1 ) a 2 = 1 + a ( n − j + j a 2 ) ( 1 − a ) ( j a 2 − a + n − j − 1 ) ≥ 1. If κ 1 ≥ 1 \kappa_{1}\ge1 κ 1 ≥ 1 (10) and (7) ,
F ≥ 1 , ϕ ≥ 1. (70) F\ge1,\qquad\phi\ge1. \tag*{(70)} F ≥ 1 , ϕ ≥ 1. (70) Moreover, setting
β n , k , ℓ = ( n − k n − ℓ ) 1 / ( k − ℓ ) > 0 , \beta_{n,k,\ell}=\left(\frac{n-k}{n-\ell}\right)^{1/(k-\ell)}>0, β n , k , ℓ = ( n − ℓ n − k ) 1/ ( k − ℓ ) > 0 , we have, whenever a = κ 1 < 1 a=\kappa_{1}<1 a = κ 1 < 1
F ≥ F ( a , a − 1 , … , a − 1 ) = 1 a ( n − k + k a 2 n − ℓ + ℓ a 2 ) 1 / ( k − ℓ ) ≥ β n , k , ℓ a . (71) F\ge F(a,a^{-1},\ldots,a^{-1})=\frac{1}{a}\left(\frac{n-k+ka^{2}}{n-\ell+\ell a^{2}}\right)^{1/(k-\ell)}\ge\frac{\beta_{n,k,\ell}}{a}. \tag*{(71)} F ≥ F ( a , a − 1 , … , a − 1 ) = a 1 ( n − ℓ + ℓ a 2 n − k + k a 2 ) 1/ ( k − ℓ ) ≥ a β n , k , ℓ . (71) Since every factor in (10) is at least one,
E k E k − 1 ≥ 1 , E ℓ + 1 E ℓ ≤ F k − ℓ . \frac{E_{k}}{E_{k-1}}\ge1,\qquad\frac{E_{\ell+1}}{E_{\ell}}\le F^{k-\ell}. E k − 1 E k ≥ 1 , E ℓ E ℓ + 1 ≤ F k − ℓ . The identities (14) therefore imply
F ˙ i j g i j ≤ k k − ℓ F , F ˙ i j ( h 2 ) i j ≤ n − ℓ k − ℓ F k − ℓ + 1 . (72) \dot{F}^{ij}g_{ij}\le\frac{k}{k-\ell}F,\qquad\dot{F}^{ij}(h^{2})_{ij}\le\frac{n-\ell}{k-\ell}F^{k-\ell+1}. \tag*{(72)} F ˙ ij g ij ≤ k − ℓ k F , F ˙ ij ( h 2 ) ij ≤ k − ℓ n − ℓ F k − ℓ + 1 . (72) Proposition 4.6. There exists a constant C F > 0 C_{F}>0 C F > 0 n n n k k k ℓ \ell ℓ M 0 M_{0} M 0 t ∈ [ 0 , T max ) t\in[0,T_{\max}) t ∈ [ 0 , T m a x )
1 ≤ F ≤ C F , 1 ≤ ϕ ( t ) ≤ C F . (73) 1\le F\le C_{F},\qquad1\le\phi(t)\le C_{F}. \tag*{(73)} 1 ≤ F ≤ C F , 1 ≤ ϕ ( t ) ≤ C F . (73) Proof. Fix t 0 < T max t_0<T_{\max} t 0 < T m a x Lemma 4.5 . On the interval [ t 0 , min { T max , t 0 + τ 0 } ) [t_0,\min\{T_{\max},t_0+\tau_0\}) [ t 0 , min { T m a x , t 0 + τ 0 }) u u u
c = 1 2 sinh ( r 0 / 4 ) , Z = F u − c . c=\frac{1}{2}\sinh(r_0/4),\qquad Z=\frac{F}{u-c}. c = 2 1 sinh ( r 0 /4 ) , Z = u − c F . By (69) ,
c ≤ u − c ≤ sinh ( 2 R 0 ) , r ≤ 2 R 0 . c\leq u-c\leq\sinh(2R_0),\qquad r\leq2R_0. c ≤ u − c ≤ sinh ( 2 R 0 ) , r ≤ 2 R 0 . At a spatial maximum of Z Z Z
∇ F = F u − c ∇ u . \nabla F=\frac{F}{u-c}\nabla u. ∇ F = u − c F ∇ u . Using (38) and (40) , the gradient terms in the quotient calculation cancel, and we obtain
( ∂ t − L ) Z = − c F + ϕ ( u − c ) ( u − c ) 2 F ˙ i j ( h 2 ) i j − F − ϕ u − c F ˙ i j g i j + F ( 2 F − ϕ ) ( u − c ) 2 cosh r . (74) (\partial_t-\mathcal{L})Z=-\frac{cF+\phi(u-c)}{(u-c)^2}\dot{F}^{ij}(h^2)_{ij}-\frac{F-\phi}{u-c}\dot{F}^{ij}g_{ij}+\frac{F(2F-\phi)}{(u-c)^2}\cosh r. \tag*{(74)} ( ∂ t − L ) Z = − ( u − c ) 2 c F + ϕ ( u − c ) F ˙ ij ( h 2 ) ij − u − c F − ϕ F ˙ ij g ij + ( u − c ) 2 F ( 2 F − ϕ ) cosh r . (74) The terms containing ϕ \phi ϕ
ϕ ( u − c ) 2 [ ( u − c ) ( F i j g i j − F ˙ i j ( h 2 ) i j ) − F cosh r ] . \frac{\phi}{(u-c)^2}\left[(u-c)\left(F^{ij}g_{ij}-\dot{F}^{ij}(h^2)_{ij}\right)-F\cosh r\right]. ( u − c ) 2 ϕ [ ( u − c ) ( F ij g ij − F ˙ ij ( h 2 ) ij ) − F cosh r ] . By (13) and (72) , this expression is nonpositive whenever F ≥ k / ( k − ℓ ) F\geq k/(k-\ell) F ≥ k / ( k − ℓ ) − F F ˙ i j g i j / ( u − c ) -F\dot{F}^{ij}g_{ij}/(u-c) − F F ˙ ij g ij / ( u − c ) Z ≥ k / ( ( k − ℓ ) c ) Z\geq k/((k-\ell)c) Z ≥ k / (( k − ℓ ) c )
( ∂ t − L ) Z ≤ − c F 3 ( u − c ) 2 + 2 cosh r F 2 ( u − c ) 2 ≤ − c 2 Z 3 + 2 cosh ( 2 R 0 ) Z 2 . (75) \begin{aligned}
(\partial_t-\mathcal{L})Z\leq-\frac{cF^3}{(u-c)^2}+2\cosh r\frac{F^2}{(u-c)^2} \\
&\leq-c^2Z^3+2\cosh(2R_0)Z^2. \tag*{(75)}
\end{aligned} ( ∂ t − L ) Z ≤ − ( u − c ) 2 c F 3 + 2 cosh r ( u − c ) 2 F 2 ≤ − c 2 Z 3 + 2 cosh ( 2 R 0 ) Z 2 . (75) Choose a constant Z ∗ > 0 Z_*>0 Z ∗ > 0 n n n k k k ℓ \ell ℓ r 0 r_0 r 0 R 0 R_0 R 0
Z ∗ ≥ k ( k − ℓ ) c , − c 2 z 3 + 2 cosh ( 2 R 0 ) z 2 ≤ − c 2 2 z 3 for z ≥ Z ∗ . Z_*\geq\frac{k}{(k-\ell)c},\qquad-c^2z^3+2\cosh(2R_0)z^2\leq-\frac{c^2}{2}z^3\quad\text{for }z\geq Z_*. Z ∗ ≥ ( k − ℓ ) c k , − c 2 z 3 + 2 cosh ( 2 R 0 ) z 2 ≤ − 2 c 2 z 3 for z ≥ Z ∗ . For t > t 0 t>t_0 t > t 0
z ( t ) = Z ∗ + 1 c t − t 0 z(t)=Z_*+\frac{1}{c\sqrt{t-t_0}} z ( t ) = Z ∗ + c t − t 0 1 satisfies
z ′ ( t ) = − c 2 2 ( z − Z ∗ ) 3 > − c 2 2 z 3 , lim t ↓ t 0 z ( t ) = + ∞ . z'(t)=-\frac{c^2}{2}(z-Z_*)^3>-\frac{c^2}{2}z^3,\qquad\lim_{t\downarrow t_0}z(t)=+\infty. z ′ ( t ) = − 2 c 2 ( z − Z ∗ ) 3 > − 2 c 2 z 3 , t ↓ t 0 lim z ( t ) = + ∞. At a first contact of Z Z Z z z z Z Z Z (75) contradicts the preceding strict inequality. The comparison principle therefore gives
Z ( x , t ) ≤ Z ∗ + 1 c t − t 0 , t 0 < t < min { T max , t 0 + τ 0 } . Z(x,t)\leq Z_*+\frac{1}{c\sqrt{t-t_0}},\qquad t_0<t<\min\{T_{\max},t_0+\tau_0\}. Z ( x , t ) ≤ Z ∗ + c t − t 0 1 , t 0 < t < min { T m a x , t 0 + τ 0 } . For each t ≥ τ 0 / 2 t\geq\tau_0/2 t ≥ τ 0 /2 t < T max t<T_{\max} t < T m a x t 0 = t − τ 0 / 2 t_0=t-\tau_0/2 t 0 = t − τ 0 /2
F ( x , t ) ≤ sinh ( 2 R 0 ) ( Z ∗ + 2 c τ 0 ) . F(x,t)\leq\sinh(2R_0)\left(Z_*+\frac{\sqrt{2}}{c\sqrt{\tau_0}}\right). F ( x , t ) ≤ sinh ( 2 R 0 ) ( Z ∗ + c τ 0 2 ) . On the initial interval, use the center chosen at t 0 = 0 t_0=0 t 0 = 0 (75) yields
Z ( x , t ) ≤ max { Z ∗ , max M 0 Z ( ⋅ , 0 ) } , 0 ≤ t < min { T max , τ 0 } . Z(x,t)\leq\max\left\{Z_*,\max_{M_0}Z(\cdot,0)\right\},\qquad0\leq t<\min\{T_{\max},\tau_0\}. Z ( x , t ) ≤ max { Z ∗ , M 0 max Z ( ⋅ , 0 ) } , 0 ≤ t < min { T m a x , τ 0 } . Since u − c ≥ c u-c\geq c u − c ≥ c u − c ≤ sinh ( 2 R 0 ) u-c\leq\sinh(2R_0) u − c ≤ sinh ( 2 R 0 ) F ≤ C F F\leq C_F F ≤ C F [ 0 , T max ) [0,T_{\max}) [ 0 , T m a x ) C F C_F C F n n n k k k ℓ \ell ℓ M 0 M_0 M 0
Finally, (7) gives
ϕ ( t ) = ∫ M t E ℓ F d μ t ∫ M t E ℓ d μ t . \phi(t)=\frac{\int_{M_t}E_\ell F\,d\mu_t}{\int_{M_t}E_\ell\,d\mu_t}. ϕ ( t ) = ∫ M t E ℓ d μ t ∫ M t E ℓ F d μ t . Thus ϕ ( t ) ≤ C F \phi(t)\le C_F ϕ ( t ) ≤ C F (70) , this proves (73) . □
Curvature estimates and long-time existence
Proposition 4.7. There exist positive constants c κ c_\kappa c κ C κ C_\kappa C κ n n n k k k ℓ \ell ℓ M 0 M_0 M 0
c κ ≤ κ i ( x , t ) ≤ C κ , i = 1 , … , n , ( x , t ) ∈ M × [ 0 , T max ) . c_\kappa\le\kappa_i(x,t)\le C_\kappa,\qquad i=1,\ldots,n,\qquad(x,t)\in M\times[0,T_{\max}). c κ ≤ κ i ( x , t ) ≤ C κ , i = 1 , … , n , ( x , t ) ∈ M × [ 0 , T m a x ) . Proof. We first prove the lower bound. At a point where a = κ 1 < 1 a=\kappa_1<1 a = κ 1 < 1 κ i ≥ a − 1 \kappa_i\ge a^{-1} κ i ≥ a − 1 i ≥ 2 i\ge2 i ≥ 2 (71) and Proposition 4.6,
C F ≥ F ≥ 1 a ( n − k + k a 2 n − ℓ + ℓ a 2 ) 1 / ( k − ℓ ) ≥ β n , k , ℓ a . C_F\ge F\ge\frac{1}{a}\left(\frac{n-k+ka^2}{n-\ell+\ell a^2}\right)^{1/(k-\ell)}\ge\frac{\beta_{n,k,\ell}}{a}. C F ≥ F ≥ a 1 ( n − ℓ + ℓ a 2 n − k + k a 2 ) 1/ ( k − ℓ ) ≥ a β n , k , ℓ . Since k ≤ n − 1 k\le n-1 k ≤ n − 1
κ 1 ≥ c κ = min { 1 , β n , k , ℓ C F } > 0. (76) \kappa_1\ge c_\kappa=\min\left\{1,\frac{\beta_{n,k,\ell}}{C_F}\right\}>0. \tag*{(76)} κ 1 ≥ c κ = min { 1 , C F β n , k , ℓ } > 0. (76) The bound also holds where κ 1 ≥ 1 \kappa_1\ge1 κ 1 ≥ 1
For the upper bound, choose C κ > max M 0 κ n C_\kappa>\max_{M_0}\kappa_n C κ > max M 0 κ n
( n − ℓ ) C F k − ℓ + 1 + k C F k − ℓ C κ − C κ 2 + C F < 0. \frac{(n-\ell)C_F^{k-\ell+1}+kC_F}{k-\ell}C_\kappa-C_\kappa^2+C_F<0. k − ℓ ( n − ℓ ) C F k − ℓ + 1 + k C F C κ − C κ 2 + C F < 0. Suppose that κ n \kappa_n κ n C κ C_\kappa C κ ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) t 0 < T max t_0<T_{\max} t 0 < T m a x e n e_n e n g ( t ) g(t) g ( t ) v v v
v = e n , ∇ i v = 0 , ∂ t v = G W ( v ) at ( x 0 , t 0 ) . v=e_n,\qquad\nabla_i v=0,\qquad\partial_t v=G\mathcal{W}(v)\quad\text{at }(x_0,t_0). v = e n , ∇ i v = 0 , ∂ t v = G W ( v ) at ( x 0 , t 0 ) . These choices are compatible with ∂ t g = − 2 G h \partial_tg=-2Gh ∂ t g = − 2 G h
f = h ( v , v ) . f=h(v,v). f = h ( v , v ) . Then f ≤ κ n f\le\kappa_n f ≤ κ n ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) f ≤ C κ f\le C_\kappa f ≤ C κ f ( x 0 , t 0 ) = C κ f(x_0,t_0)=C_\kappa f ( x 0 , t 0 ) = C κ
∂ t f ≥ 0 , ∇ 2 f ≤ 0 , ( ∂ t − L ) f ≥ 0 at ( x 0 , t 0 ) . \partial_t f\ge0,\qquad\nabla^2f\le0,\qquad(\partial_t-\mathcal{L})f\ge0\quad\text{at }(x_0,t_0). ∂ t f ≥ 0 , ∇ 2 f ≤ 0 , ( ∂ t − L ) f ≥ 0 at ( x 0 , t 0 ) . All subsequent calculations are made at the contact point. Since v v v ∇ v = 0 \nabla v=0 ∇ v = 0 W ( v ) = κ n v \mathcal{W}(v)=\kappa_nv W ( v ) = κ n v
L f = F ˙ i j ∇ i ∇ j h n n , ∂ t f = ∂ t h n n + 2 G κ n 2 = ∂ t h n n . \mathcal{L}f=\dot{F}^{ij}\nabla_i\nabla_jh_{nn},\qquad\partial_tf=\partial_th_{nn}+2G\kappa_n^2=\partial_th^n_n. L f = F ˙ ij ∇ i ∇ j h nn , ∂ t f = ∂ t h nn + 2 G κ n 2 = ∂ t h n n . Here the time derivatives of the tensor components are taken in fixed local coordinates agreeing with the principal frame at ( x 0 , t 0 ) (x_0,t_0) ( x 0 , t 0 ) (37) , we obtain
( ∂ t − L ) f = F ¨ i j , p q ∇ n h i j ∇ n h p q + F ˙ i j ( ( h 2 ) i j + g i j ) f − ϕ f 2 + ϕ − 2 F . (\partial_t-\mathcal{L})f=\ddot{F}^{ij,pq}\nabla_nh_{ij}\nabla_nh_{pq}+\dot{F}^{ij}\bigl((h^2)_{ij}+g_{ij}\bigr)f-\phi f^2+\phi-2F. ( ∂ t − L ) f = F ¨ ij , pq ∇ n h ij ∇ n h pq + F ˙ ij ( ( h 2 ) ij + g ij ) f − ϕ f 2 + ϕ − 2 F . The function F F F
F ¨ i j , p q ∇ n h i j ∇ n h p q ≤ 0. \ddot{F}^{ij,pq}\nabla_nh_{ij}\nabla_nh_{pq}\le0. F ¨ ij , pq ∇ n h ij ∇ n h pq ≤ 0. Moreover, (72) and (73) imply
F ˙ i j ( h 2 ) i j ≤ n − ℓ k − ℓ C F k − ℓ + 1 , F ˙ i j g i j ≤ k k − ℓ C F , 1 ≤ ϕ ≤ C F . \dot{F}^{ij}(h^2)_{ij}\le\frac{n-\ell}{k-\ell}C_F^{k-\ell+1},\qquad\dot{F}^{ij}g_{ij}\le\frac{k}{k-\ell}C_F,\qquad1\le\phi\le C_F. F ˙ ij ( h 2 ) ij ≤ k − ℓ n − ℓ C F k − ℓ + 1 , F ˙ ij g ij ≤ k − ℓ k C F , 1 ≤ ϕ ≤ C F . Consequently,
0 ≤ ( ∂ t − L ) f ≤ ( n − ℓ ) C F k − ℓ + 1 + k C F k − ℓ f − f 2 + C F = ( n − ℓ ) C F k − ℓ + 1 + k C F k − ℓ C κ − C κ 2 + C F < 0 , (77) 0 \leq(\partial_t-\mathcal{L})f
\leq\frac{(n-\ell)C_F^{k-\ell+1}+kC_F}{k-\ell}f-f^2+C_F
=\frac{(n-\ell)C_F^{k-\ell+1}+kC_F}{k-\ell}C_\kappa-C_\kappa^2+C_F<0, \tag*{(77)} 0 ≤ ( ∂ t − L ) f ≤ k − ℓ ( n − ℓ ) C F k − ℓ + 1 + k C F f − f 2 + C F = k − ℓ ( n − ℓ ) C F k − ℓ + 1 + k C F C κ − C κ 2 + C F < 0 , (77) a contradiction. Hence κ n < C κ \kappa_n<C_\kappa κ n < C κ [ 0 , T max ) [0,T_{\max}) [ 0 , T m a x ) (76) , this completes the proof. □
We have obtained uniform two-sided bounds for the principal curvatures. On each time interval in Lemma 4.5 , write M t M_t M t r = r ( θ , t ) r=r(\theta,t) r = r ( θ , t ) (69) and (20) give
r 0 4 ≤ r ≤ 2 R 0 , v = sinh r u ≤ C , ∣ D r ∣ σ + ∣ D 2 r ∣ σ ≤ C . \frac{r_0}{4}\leq r\leq2R_0,\qquad v=\frac{\sinh r}{u}\leq C,\qquad|Dr|_\sigma+|D^2r|_\sigma\leq C. 4 r 0 ≤ r ≤ 2 R 0 , v = u sinh r ≤ C , ∣ D r ∣ σ + ∣ D 2 r ∣ σ ≤ C . Thus the corresponding radial graphs have uniform C 2 C^2 C 2 (10) and (11) , F F F [6 Section 5] , as in [2 Section 4.3] , to obtain uniform Hölder estimates for the second derivatives. Parabolic Schauder estimates [24 ] then give all higher-order bounds. A standard continuation argument yields T max = ∞ T_{\max}=\infty T m a x = ∞
Convergence and the inequalities
By (30) and positivity of W k W_k W k
∫ 0 ∞ ∫ M t E ℓ ( F k − ℓ − ϕ k − ℓ ) ( F − ϕ ) d μ t d t < ∞ . \int_0^\infty\int_{M_t}E_\ell(F^{k-\ell}-\phi^{k-\ell})(F-\phi)\,d\mu_t\,dt<\infty. ∫ 0 ∞ ∫ M t E ℓ ( F k − ℓ − ϕ k − ℓ ) ( F − ϕ ) d μ t d t < ∞. Since F , ϕ ≥ 1 F,\phi\geq1 F , ϕ ≥ 1
( F k − ℓ − ϕ k − ℓ ) ( F − ϕ ) = ( F − ϕ ) 2 ∑ j = 0 k − ℓ − 1 F k − ℓ − 1 − j ϕ j ≥ ( k − ℓ ) ( F − ϕ ) 2 . (F^{k-\ell}-\phi^{k-\ell})(F-\phi)=(F-\phi)^2\sum_{j=0}^{k-\ell-1}F^{k-\ell-1-j}\phi^j\geq(k-\ell)(F-\phi)^2. ( F k − ℓ − ϕ k − ℓ ) ( F − ϕ ) = ( F − ϕ ) 2 j = 0 ∑ k − ℓ − 1 F k − ℓ − 1 − j ϕ j ≥ ( k − ℓ ) ( F − ϕ ) 2 . Hence there exist times t j → ∞ t_j\to\infty t j → ∞
∫ M t j E ℓ ( F − ϕ ) 2 d μ t j ⟶ 0. (78) \int_{M_{t_j}}E_\ell(F-\phi)^2\,d\mu_{t_j}\longrightarrow0. \tag*{(78)} ∫ M t j E ℓ ( F − ϕ ) 2 d μ t j ⟶ 0. (78) The uniform curvature and higher-order estimates, together with the fixed outer ball and the inner-radius bound, imply that, after passing to a subsequence and reparametrizing, M t j M_{t_j} M t j M ∞ M_\infty M ∞ (73) , we may also assume that ϕ ( t j ) → c > 0 \phi(t_j)\to c>0 ϕ ( t j ) → c > 0 (78) gives
∫ M ∞ E ℓ ( F − c ) 2 d μ ∞ = 0. \int_{M_\infty}E_\ell(F-c)^2\,d\mu_\infty=0. ∫ M ∞ E ℓ ( F − c ) 2 d μ ∞ = 0. Since E ℓ > 0 E_\ell>0 E ℓ > 0 E k = c k − ℓ E ℓ E_k=c^{k-\ell}E_\ell E k = c k − ℓ E ℓ M ∞ M_\infty M ∞ [21 Theorem B] implies that it is a geodesic sphere. Its radius r ∞ r_\infty r ∞ W ℓ ( B r ∞ ) = W ℓ ( Ω 0 ) W_\ell(B_{r_\infty})=W_\ell(\Omega_0) W ℓ ( B r ∞ ) = W ℓ ( Ω 0 )
To obtain convergence of the entire flow, we apply the linearization argument in [6 Section 6] , as used in [2 Section 4.3] . At a geodesic sphere, symmetry and homogeneity give F ˙ i = 1 / n \dot F^i=1/n F ˙ i = 1/ n F F F E ℓ d μ E_\ell\,d\mu E ℓ d μ (7) . Consequently, δ ϕ = δ F \delta\phi=\delta F δ ϕ = δ F r ∞ r_\infty r ∞
L η = 1 n sinh 2 r ∞ ( Δ σ η + n η − n η ˉ ) , η ˉ = 1 ∣ S n ∣ ∫ S n η d σ . (79) \mathcal{L}\eta=\frac{1}{n\sinh^2 r_\infty}(\Delta_\sigma\eta+n\eta-n\bar\eta),\qquad\bar\eta=\frac{1}{|S^n|}\int_{S^n}\eta\,d\sigma. \tag*{(79)} L η = n sinh 2 r ∞ 1 ( Δ σ η + n η − n η ˉ ) , η ˉ = ∣ S n ∣ 1 ∫ S n η d σ . (79) The nearby equilibria are precisely the geodesic spheres by [21 Theorem B] . They form an ( n + 2 ) (n+2) ( n + 2 ) ker L \ker\mathcal{L} ker L j ≥ 2 j \ge2 j ≥ 2
L η = n − j ( j + n − 1 ) n sinh 2 r ∞ η , n − j ( j + n − 1 ) < 0. \mathcal{L}\eta= \frac{n-j(j+n-1)}{n\sinh^{2}r_{\infty}}\eta,\qquad n-j(j+n-1)<0. L η = n sinh 2 r ∞ n − j ( j + n − 1 ) η , n − j ( j + n − 1 ) < 0. The normal-graph operator is smooth and uniformly parabolic near the sphere. Thus the normal stability theorem [29 Theorem 3.1] , applied with A = − L A=-\mathcal{L} A = − L M ∞ M_{\infty} M ∞ M t j → M ∞ M_{t_j}\to M_{\infty} M t j → M ∞ r ∞ r_{\infty} r ∞ W ℓ W_{\ell} W ℓ Theorem 1.3 .
Proof of Theorem 1.1 . First suppose that ∂ Ω \partial\Omega ∂ Ω 1 ≤ ℓ < k 1\le\ell<k 1 ≤ ℓ < k k + ℓ ≤ n k+\ell\le n k + ℓ ≤ n (30) and Theorem 1.3 give
W k ( Ω ) ≥ W k ( B r ∞ ) = f k ∘ f ℓ − 1 ( W ℓ ( Ω ) ) . W_k(\Omega)\ge W_k(B_{r_\infty})=f_k\circ f_\ell^{-1}(W_\ell(\Omega)). W k ( Ω ) ≥ W k ( B r ∞ ) = f k ∘ f ℓ − 1 ( W ℓ ( Ω )) . For ℓ = 0 \ell=0 ℓ = 0 [2 Corollary 1.5] .
Now assume sec ∂ Ω ≥ 0 \sec_{\partial\Omega}\ge0 sec ∂ Ω ≥ 0 (17) give
κ i ≥ 0 , κ i κ j ≥ 1 ( i ≠ j ) , hence κ i > 0. \kappa_i\ge0,\qquad\kappa_i\kappa_j\ge1\quad(i\ne j),\qquad\text{hence }\kappa_i>0. κ i ≥ 0 , κ i κ j ≥ 1 ( i = j ) , hence κ i > 0. Let Ω c \Omega_c Ω c Lemma 2.1 ,
sec ∂ Ω c > 0 , ∂ Ω c ⟶ ∂ Ω smoothly as c ↑ 1. \sec_{\partial\Omega_c}>0,\qquad\partial\Omega_c\longrightarrow\partial\Omega\quad\text{smoothly as }c\uparrow1. sec ∂ Ω c > 0 , ∂ Ω c ⟶ ∂ Ω smoothly as c ↑ 1. Applying the positive-curvature case and passing to the limit yields
W k ( Ω ) = lim c ↑ 1 W k ( Ω c ) ≥ lim c ↑ 1 f k ∘ f ℓ − 1 ( W ℓ ( Ω c ) ) = f k ∘ f ℓ − 1 ( W ℓ ( Ω ) ) . (80) \begin{aligned}
W_k(\Omega)&=\lim_{c\uparrow1}W_k(\Omega_c)\\
&\ge\lim_{c\uparrow1}f_k\circ f_\ell^{-1}(W_\ell(\Omega_c))
=f_k\circ f_\ell^{-1}(W_\ell(\Omega)). \tag*{(80)}
\end{aligned} W k ( Ω ) = c ↑ 1 lim W k ( Ω c ) ≥ c ↑ 1 lim f k ∘ f ℓ − 1 ( W ℓ ( Ω c )) = f k ∘ f ℓ − 1 ( W ℓ ( Ω )) . (80) Suppose equality holds. Starting from ∂ Ω \partial\Omega ∂ Ω (6) –(7) . For ℓ = 0 \ell=0 ℓ = 0 F = E k 1 / k F=E_k^{1/k} F = E k 1/ k E 0 = 1 E_0=1 E 0 = 1 § 2 gives a common interval [ 0 , T ] [0,T] [ 0 , T ] ∂ Ω c \partial\Omega_c ∂ Ω c c ∈ [ c 0 , 1 ] c\in[c_0,1] c ∈ [ c 0 , 1 ] c 0 < 1 c_0<1 c 0 < 1
M c , t ⟶ M t = ∂ Ω t in C 2 as c ↑ 1 , Ω 0 = Ω . M_{c,t}\longrightarrow M_t=\partial\Omega_t\quad\text{in }C^2\text{ as }c\uparrow1,\qquad\Omega_0=\Omega. M c , t ⟶ M t = ∂ Ω t in C 2 as c ↑ 1 , Ω 0 = Ω. For c ∈ [ c 0 , 1 ) c\in[c_0,1) c ∈ [ c 0 , 1 ) Theorem 3.1 when ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 [2 Theorem 3.1] when ℓ = 0 \ell=0 ℓ = 0 sec M t ≥ 0 \sec_{M_t}\ge0 sec M t ≥ 0 [ 0 , T ] [0,T] [ 0 , T ] (30) also hold for ℓ = 0 \ell=0 ℓ = 0 (80) gives
W k ( Ω 0 ) ≥ W k ( Ω t ) ≥ f k ∘ f ℓ − 1 ( W ℓ ( Ω t ) ) = f k ∘ f ℓ − 1 ( W ℓ ( Ω 0 ) ) = W k ( Ω 0 ) . \begin{aligned}
W_k(\Omega_0)&\ge W_k(\Omega_t)\ge f_k\circ f_\ell^{-1}(W_\ell(\Omega_t))\\
&=f_k\circ f_\ell^{-1}(W_\ell(\Omega_0))=W_k(\Omega_0).
\end{aligned} W k ( Ω 0 ) ≥ W k ( Ω t ) ≥ f k ∘ f ℓ − 1 ( W ℓ ( Ω t )) = f k ∘ f ℓ − 1 ( W ℓ ( Ω 0 )) = W k ( Ω 0 ) . Thus W k ( Ω t ) W_k(\Omega_t) W k ( Ω t ) (30) at t = 0 t=0 t = 0 E ℓ > 0 E_\ell>0 E ℓ > 0
F ≡ ϕ ( 0 ) , E k = ϕ ( 0 ) k − ℓ E ℓ on ∂ Ω . F\equiv\phi(0),\qquad E_k=\phi(0)^{k-\ell}E_\ell\quad\text{on }\partial\Omega. F ≡ ϕ ( 0 ) , E k = ϕ ( 0 ) k − ℓ E ℓ on ∂ Ω. Koh’s theorem [21 Theorem B] implies that Ω \Omega Ω
Proof of Corollary 1.2. Convexity and nonnegative sectional curvature imply strict convexity, as in the proof of Theorem 1.1 . For 3 ≤ n ≤ 5 3\le n\le5 3 ≤ n ≤ 5 0 ≤ ℓ < k ≤ n − 2 0\le\ell<k\le n-2 0 ≤ ℓ < k ≤ n − 2 k + ℓ ≤ 2 n − 5 ≤ n k+\ell\le2n-5\le n k + ℓ ≤ 2 n − 5 ≤ n Theorem 1.1 gives these comparisons. The comparisons with upper index n − 1 n-1 n − 1 [17 Theorem 1.3] , and those with upper index n n n [7 ] . The latter result is also stated in [15 Theorem 4.2(2)] . These two endpoint results also give all comparisons when n = 2 n=2 n = 2
Sharpness of the index range
We prove Theorem 1.4 in three steps. First, we construct a strictly convex local graph in the Klein model whose minimum sectional curvature is nonnegative, vanishes only at the origin, and has a negative initial time derivative there. We then complete the graph to a smooth closed hypersurface with nonnegative sectional curvature, leaving a neighbourhood of the origin unchanged. Finally, Klein contraction gives strictly positive initial sectional curvature, and short-time continuous dependence yields the required loss of positivity.
Proof of Theorem 1.4. Step 1. Construction of the local graph. For a graph Y = ( x , w ( x ) ) Y=(x,w(x)) Y = ( x , w ( x )) g = ( g i j ) g=(g_{ij}) g = ( g ij ) h = ( h i j ) h=(h_{ij}) h = ( h ij ) D E D_{E} D E
A = g − 1 / 2 h g − 1 / 2 . (81) \mathcal{A}=g^{-1/2}hg^{-1/2}. \tag*{(81)} A = g − 1/2 h g − 1/2 . (81) By (22) and (23) , this matrix is a positive congruence of D E 2 w D_{E}^{2}w D E 2 w ( x , w , D E w ) (x,w,D_{E}w) ( x , w , D E w ) w ( o ) = D E w ( o ) = 0 w(o)=D_{E}w(o)=0 w ( o ) = D E w ( o ) = 0 o = 0 o=0 o = 0
g ( o ) = I , D E g ( o ) = 0 , A ( o ) = D E 2 w ( o ) , ∇ m h i j ( o ) = ∂ m i j w ( o ) . g(o)=I,\qquad D_{E}g(o)=0,\qquad\mathcal{A}(o)=D_{E}^{2}w(o),\qquad\nabla_{m}h_{ij}(o)=\partial_{mij}w(o). g ( o ) = I , D E g ( o ) = 0 , A ( o ) = D E 2 w ( o ) , ∇ m h ij ( o ) = ∂ mij w ( o ) . Write x = ( x 1 , x 2 , y ) x=(x_{1},x_{2},y) x = ( x 1 , x 2 , y ) y ∈ R n − 2 y\in\mathbb{R}^{n-2} y ∈ R n − 2 a > 1 a>1 a > 1
w a ( x ) = 1 2 ( a − 1 x 1 2 + a x 2 2 + a 2 ∣ y ∣ 2 ) + x 1 3 6 − a 2 x 1 x 2 2 2 + a 3 4 ∣ x ∣ 4 + 1 12 ( a − 1 x 1 4 + a x 2 4 ) . (82) w_{a}(x)=\frac{1}{2}\left(a^{-1}x_{1}^{2}+ax_{2}^{2}+a^{2}|y|^{2}\right)+\frac{x_{1}^{3}}{6}-\frac{a^{2}x_{1}x_{2}^{2}}{2}+\frac{a^{3}}{4}|x|^{4}+\frac{1}{12}\left(a^{-1}x_{1}^{4}+ax_{2}^{4}\right). \tag*{(82)} w a ( x ) = 2 1 ( a − 1 x 1 2 + a x 2 2 + a 2 ∣ y ∣ 2 ) + 6 x 1 3 − 2 a 2 x 1 x 2 2 + 4 a 3 ∣ x ∣ 4 + 12 1 ( a − 1 x 1 4 + a x 2 4 ) . (82) Let A a \mathcal{A}_{a} A a (81) of this graph. At o o o
A a ( o ) = diag ( a − 1 , a , a 2 , … , a 2 ) , \mathcal{A}_{a}(o)=\operatorname{diag}(a^{-1},a,a^{2},\ldots,a^{2}), A a ( o ) = diag ( a − 1 , a , a 2 , … , a 2 ) , and the only nonzero components of ∇ h \nabla h ∇ h
∇ 1 h 11 ( o ) = 1 , ∇ 1 h 22 ( o ) = ∇ 2 h 12 ( o ) = ∇ 2 h 21 ( o ) = − a 2 . (83) \nabla_{1}h_{11}(o)=1,\qquad\nabla_{1}h_{22}(o)=\nabla_{2}h_{12}(o)=\nabla_{2}h_{21}(o)=-a^{2}. \tag*{(83)} ∇ 1 h 11 ( o ) = 1 , ∇ 1 h 22 ( o ) = ∇ 2 h 12 ( o ) = ∇ 2 h 21 ( o ) = − a 2 . (83) The first two principal curvatures are simple at o o o S a = κ 1 κ 2 − 1 S_{a}=\kappa_{1}\kappa_{2}-1 S a = κ 1 κ 2 − 1 o o o
S a ( o ) = 0 , ∇ S a ( o ) = 0. S_{a}(o)=0,\qquad\nabla S_{a}(o)=0. S a ( o ) = 0 , ∇ S a ( o ) = 0. We compute the quadratic part of S a S_{a} S a
g = I + ∣ x ∣ 2 I + x ⊗ x + D E w a ⊗ D E w a + O a ( ∣ x ∣ 3 ) , h = ( 1 + ∣ x ∣ 2 − ∣ D E w a ∣ 2 2 ) D E 2 w a + O a ( ∣ x ∣ 3 ) , g − 1 / 2 = I − 1 2 ( ∣ x ∣ 2 I + x ⊗ x + D E w a ⊗ D E w a ) + O a ( ∣ x ∣ 3 ) . (84) \begin{aligned}
g&=I+|x|^{2}I+x\otimes x+D_{E}w_{a}\otimes D_{E}w_{a}+O_{a}(|x|^{3}),\\
h&=\left(1+\frac{|x|^{2}-|D_{E}w_{a}|^{2}}{2}\right)D_{E}^{2}w_{a}+O_{a}(|x|^{3}),\\
g^{-1/2}&=I-\frac{1}{2}\left(|x|^{2}I+x\otimes x+D_{E}w_{a}\otimes D_{E}w_{a}\right)+O_{a}(|x|^{3}). \tag*{(84)}
\end{aligned} g h g − 1/2 = I + ∣ x ∣ 2 I + x ⊗ x + D E w a ⊗ D E w a + O a ( ∣ x ∣ 3 ) , = ( 1 + 2 ∣ x ∣ 2 − ∣ D E w a ∣ 2 ) D E 2 w a + O a ( ∣ x ∣ 3 ) , = I − 2 1 ( ∣ x ∣ 2 I + x ⊗ x + D E w a ⊗ D E w a ) + O a ( ∣ x ∣ 3 ) . (84) Substitution into (81) yields, for i = 1 , 2 i=1,2 i = 1 , 2
( A a ) i i = ∂ i i w a − κ i ( o ) 2 ∑ j = 1 n ( 1 + κ j ( o ) 2 ) x j 2 − κ i ( o ) ( 1 + κ i ( o ) 2 ) x i 2 + O a ( ∣ x ∣ 3 ) . (85) (\mathcal{A}_{a})_{ii}=\partial_{ii}w_{a}-\frac{\kappa_{i}(o)}{2}\sum_{j=1}^{n}(1+\kappa_{j}(o)^{2})x_{j}^{2}-\kappa_{i}(o)(1+\kappa_{i}(o)^{2})x_{i}^{2}+O_{a}(|x|^{3}). \tag*{(85)} ( A a ) ii = ∂ ii w a − 2 κ i ( o ) j = 1 ∑ n ( 1 + κ j ( o ) 2 ) x j 2 − κ i ( o ) ( 1 + κ i ( o ) 2 ) x i 2 + O a ( ∣ x ∣ 3 ) . (85) Moreover,
( A a ) 12 = − a 2 x 2 + O a ( ∣ x ∣ 2 ) , ( A a ) i p = O a ( ∣ x ∣ 2 ) for i = 1 , 2 , p ≥ 3. (\mathcal{A}_{a})_{12}=-a^{2}x_{2}+O_{a}(|x|^{2}),\qquad(\mathcal{A}_{a})_{ip}=O_{a}(|x|^{2})\quad\text{for }i=1,2,\quad p\geq3. ( A a ) 12 = − a 2 x 2 + O a ( ∣ x ∣ 2 ) , ( A a ) i p = O a ( ∣ x ∣ 2 ) for i = 1 , 2 , p ≥ 3. For fixed a > 1 a>1 a > 1 o o o 2 × 2 2 \times2 2 × 2 (82) and (85) , we obtain
S a ( x ) = ( A a ) 11 ( A a ) 22 − ( A a ) 12 2 − 1 + O a ( ∣ x ∣ 3 ) = ( 3 a 4 − 1 − 2 a − 2 ) x 1 2 + ( a 2 − 1 ) ( x 2 2 + ∣ y ∣ 2 ) + O a ( ∣ x ∣ 3 ) = ( a 2 − 1 ) [ ( 3 a 2 + 3 + 2 a − 2 ) x 1 2 + x 2 2 + ∣ y ∣ 2 ] + O a ( ∣ x ∣ 3 ) . (86) \begin{aligned}
S_a(x) &= (\mathcal{A}_a)_{11}(\mathcal{A}_a)_{22}-(\mathcal{A}_a)_{12}^2-1+O_a(|x|^3)\\
&=(3a^4-1-2a^{-2})x_1^2+(a^2-1)(x_2^2+|y|^2)+O_a(|x|^3)\\
&=(a^2-1)\left[(3a^2+3+2a^{-2})x_1^2+x_2^2+|y|^2\right]+O_a(|x|^3). \tag*{(86)}
\end{aligned} S a ( x ) = ( A a ) 11 ( A a ) 22 − ( A a ) 12 2 − 1 + O a ( ∣ x ∣ 3 ) = ( 3 a 4 − 1 − 2 a − 2 ) x 1 2 + ( a 2 − 1 ) ( x 2 2 + ∣ y ∣ 2 ) + O a ( ∣ x ∣ 3 ) = ( a 2 − 1 ) [ ( 3 a 2 + 3 + 2 a − 2 ) x 1 2 + x 2 2 + ∣ y ∣ 2 ] + O a ( ∣ x ∣ 3 ) . (86) Thus S a S_a S a o o o
We next compute its initial variation under the normal velocity ( ϕ − F ) ν (\phi-F)\nu ( ϕ − F ) ν (35) and (34) give, at o o o
∂ t S a ( o , 0 ) = a ∇ 11 F + a − 1 ∇ 22 F , (87) \partial_tS_a(o,0)=a\nabla_{11}F+a^{-1}\nabla_{22}F, \tag*{(87)} ∂ t S a ( o , 0 ) = a ∇ 11 F + a − 1 ∇ 22 F , (87) since the zero-order term is ( F − ϕ ) ( a + a − 1 ) ( κ 1 κ 2 − 1 ) = 0 (F-\phi)(a+a^{-1})(\kappa_1\kappa_2-1)=0 ( F − ϕ ) ( a + a − 1 ) ( κ 1 κ 2 − 1 ) = 0 ϕ ( 0 ) \phi(0) ϕ ( 0 )
At a = 1 a=1 a = 1 (84) and the identity
D E 2 ( ∣ x ∣ 4 4 ) = ∣ x ∣ 2 I + 2 x ⊗ x D_E^2\left(\frac{|x|^4}{4}\right)=|x|^2I+2x\otimes x D E 2 ( 4 ∣ x ∣ 4 ) = ∣ x ∣ 2 I + 2 x ⊗ x give
A 1 = I + ( ( x 1 + x 1 2 − x 2 − x 2 − x 1 + x 2 2 ) ⊕ 0 n − 2 ) + O ( ∣ x ∣ 3 ) . (88) \mathcal{A}_1=I+\left(\begin{pmatrix}x_1+x_1^2&-x_2\\-x_2&-x_1+x_2^2\end{pmatrix}\oplus0_{n-2}\right)+O(|x|^3). \tag*{(88)} A 1 = I + ( ( x 1 + x 1 2 − x 2 − x 2 − x 1 + x 2 2 ) ⊕ 0 n − 2 ) + O ( ∣ x ∣ 3 ) . (88) Consequently, for m = 1 , 2 m=1,2 m = 1 , 2
tr ( ∂ m A 1 ( o ) ) = 0 , tr ( ( ∂ m A 1 ( o ) ) 2 ) = 2 , tr ( ∂ m m A 1 ( o ) ) = 2. \operatorname{tr}(\partial_m\mathcal{A}_1(o))=0,\qquad\operatorname{tr}\left((\partial_m\mathcal{A}_1(o))^2\right)=2,\qquad\operatorname{tr}(\partial_{mm}\mathcal{A}_1(o))=2. tr ( ∂ m A 1 ( o )) = 0 , tr ( ( ∂ m A 1 ( o ) ) 2 ) = 2 , tr ( ∂ mm A 1 ( o )) = 2. For every symmetric matrix ξ \xi ξ F = ( E k / E ℓ ) 1 / ( k − ℓ ) F=(E_k/E_\ell)^{1/(k-\ell)} F = ( E k / E ℓ ) 1/ ( k − ℓ ) I I I
F ˙ i j ( I ) ξ i j = tr ξ n , \dot{F}^{ij}(I)\xi_{ij}=\frac{\operatorname{tr}\xi}{n}, F ˙ ij ( I ) ξ ij = n tr ξ , F ¨ i j , p q ( I ) ξ i j ξ p q = k + ℓ − 1 n 2 ( n − 1 ) ( ( tr ξ ) 2 − n tr ( ξ 2 ) ) . (89) \ddot{F}^{ij,pq}(I)\xi_{ij}\xi_{pq}=\frac{k+\ell-1}{n^2(n-1)}\left((\operatorname{tr}\xi)^2-n\operatorname{tr}(\xi^2)\right). \tag*{(89)} F ¨ ij , pq ( I ) ξ ij ξ pq = n 2 ( n − 1 ) k + ℓ − 1 ( ( tr ξ ) 2 − n tr ( ξ 2 ) ) . (89) The chain rule therefore gives
∂ m m ( F ( A 1 ) ) ∣ o = 2 n − 2 ( k + ℓ − 1 ) n ( n − 1 ) , m = 1 , 2. (90) \left.\partial_{mm}(F(\mathcal{A}_1))\right|_o=\frac{2}{n}-\frac{2(k+\ell-1)}{n(n-1)},\qquad m=1,2. \tag*{(90)} ∂ mm ( F ( A 1 )) ∣ o = n 2 − n ( n − 1 ) 2 ( k + ℓ − 1 ) , m = 1 , 2. (90) Since F F F I I I o o o a a a (87) that
lim a ↓ 1 ∂ t S a ( o , 0 ) = 4 n − 4 ( k + ℓ − 1 ) n ( n − 1 ) = − 4 ( k + ℓ − n ) n ( n − 1 ) < 0. (91) \lim_{a\downarrow1}\partial_tS_a(o,0)=\frac{4}{n}-\frac{4(k+\ell-1)}{n(n-1)}=-\frac{4(k+\ell-n)}{n(n-1)}<0. \tag*{(91)} a ↓ 1 lim ∂ t S a ( o , 0 ) = n 4 − n ( n − 1 ) 4 ( k + ℓ − 1 ) = − n ( n − 1 ) 4 ( k + ℓ − n ) < 0. (91) By (91) , choose and fix 1 < a < 2 1<a<\sqrt{2} 1 < a < 2 1 1 1
∂ t S a ( o , 0 ) < − 2 ( k + ℓ − n ) n ( n − 1 ) < 0. (92) \partial_tS_a(o,0)<-\frac{2(k+\ell-n)}{n(n-1)}<0. \tag*{(92)} ∂ t S a ( o , 0 ) < − n ( n − 1 ) 2 ( k + ℓ − n ) < 0. (92) Set w = w a w=w_a w = w a A = A a \mathcal{A}=\mathcal{A}_a A = A a (86) , there is 0 < ϱ < 1 / 16 0<\varrho<1/16 0 < ϱ < 1/16
0 < D E 2 w < 2 I , S a ( x ) ≥ a 2 − 1 2 ∣ x ∣ 2 on B ϱ ( 0 ) . (93) 0<D_E^2w<2I,\qquad S_a(x)\geq\frac{a^2-1}{2}|x|^2\quad\text{on }B_\varrho(0). \tag*{(93)} 0 < D E 2 w < 2 I , S a ( x ) ≥ 2 a 2 − 1 ∣ x ∣ 2 on B ϱ ( 0 ) . (93) We keep a a a ϱ \varrho ϱ
Step 2. Completion to a closed hypersurface. We adapt the spherical completion of [5 Sections 4–5] . For δ > 0 \delta>0 δ > 0
b δ ( x ) = 1 4 − δ − 1 16 − ∣ x ∣ 2 b_\delta(x)=\frac14-\delta-\sqrt{\frac1{16}-|x|^2} b δ ( x ) = 4 1 − δ − 16 1 − ∣ x ∣ 2 be the lower cap of the Euclidean sphere of radius 1 / 4 1/4 1/4 ( 0 , 1 / 4 − δ ) (0,1/4-\delta) ( 0 , 1/4 − δ ) B 1 / 2 ( 0 ) B_{1/2}(0) B 1/2 ( 0 )
κ i b ≥ 4 ( 1 − ∣ Y ∣ 2 ) 2 ≥ 9 4 . \kappa_i^b \ge4(1-|Y|^2)^2 \ge\frac{9}{4}. κ i b ≥ 4 ( 1 − ∣ Y ∣ 2 ) 2 ≥ 4 9 . by (22) and (23) . Since D E 2 b δ ≥ 4 I D_E^2b_\delta\ge4I D E 2 b δ ≥ 4 I (93) gives
− δ + ∣ x ∣ 2 ≤ b δ − w ≤ − δ + C ∣ x ∣ 2 on B ϱ ( 0 ) . -\delta+|x|^2\le b_\delta-w\le-\delta+C|x|^2\quad\text{on }B_\varrho(0). − δ + ∣ x ∣ 2 ≤ b δ − w ≤ − δ + C ∣ x ∣ 2 on B ϱ ( 0 ) . Consequently, w > b δ w>b_\delta w > b δ o o o b δ > w b_\delta>w b δ > w ∂ B ϱ ( 0 ) \partial B_\varrho(0) ∂ B ϱ ( 0 ) δ \delta δ Figure 1 .
Figure 1. Schematic section of the local graph w w w b δ b_\delta b δ o o o w > b δ w>b_\delta w > b δ b δ > w b_\delta>w b δ > w
The upper envelope max { w , b δ } \max\{w,b_\delta\} max { w , b δ } w w w o o o b δ b_\delta b δ χ ∈ C c ∞ ( ( − 1 , 1 ) ) \chi\in C^\infty_c((-1,1)) χ ∈ C c ∞ (( − 1 , 1 )) ∫ R χ = 1 \int_{\mathbb{R}}\chi=1 ∫ R χ = 1
ζ δ ( s ) = ∫ R ( s − δ 2 t ) + χ ( t ) d t , ( q ) + = max { q , 0 } . (94) \zeta_\delta(s)=\int_{\mathbb{R}}(s-\delta^2t)_+\chi(t)\,dt,\qquad(q)_+=\max\{q,0\}. \tag*{(94)} ζ δ ( s ) = ∫ R ( s − δ 2 t ) + χ ( t ) d t , ( q ) + = max { q , 0 } . (94) Evenness of χ \chi χ
0 ≤ ζ δ ′ ≤ 1 , ζ δ ′ ′ ≥ 0 , ζ δ ( s ) = { 0 , s ≤ − δ 2 , s , s ≥ δ 2 . 0\le\zeta_\delta'\le1,\qquad\zeta_\delta''\ge0,\qquad\zeta_\delta(s)=
\begin{cases}
0,&s\le-\delta^2,\\
s,&s\ge\delta^2.
\end{cases} 0 ≤ ζ δ ′ ≤ 1 , ζ δ ′′ ≥ 0 , ζ δ ( s ) = { 0 , s , s ≤ − δ 2 , s ≥ δ 2 . Set
w ~ = w + ζ δ ( b δ − w ) , ϑ = ζ δ ′ ( b δ − w ) . \widetilde w=w+\zeta_\delta(b_\delta-w),\qquad\vartheta=\zeta_\delta'(b_\delta-w). w = w + ζ δ ( b δ − w ) , ϑ = ζ δ ′ ( b δ − w ) . Then w ~ = w \widetilde w=w w = w o o o w ~ = b δ \widetilde w=b_\delta w = b δ ∣ b δ − w ∣ ≤ δ 2 |b_\delta-w|\le\delta^2 ∣ b δ − w ∣ ≤ δ 2 ∣ x ∣ ≤ C δ |x|\le C\sqrt{\delta} ∣ x ∣ ≤ C δ
0 ≤ w ~ − w ≤ 2 δ 2 ϑ , ∣ D E w ~ − D E w ∣ ≤ C ϑ δ . 0\le\widetilde w-w\le2\delta^2\vartheta,\qquad|D_E\widetilde w-D_Ew|\le C\vartheta\sqrt{\delta}. 0 ≤ w − w ≤ 2 δ 2 ϑ , ∣ D E w − D E w ∣ ≤ C ϑ δ . Since ζ δ ′ ′ ≥ 0 \zeta_\delta''\ge0 ζ δ ′′ ≥ 0 D E 2 b δ − D E 2 w ≥ 2 I D_E^2b_\delta-D_E^2w\ge2I D E 2 b δ − D E 2 w ≥ 2 I
D E 2 w ~ = ( 1 − ϑ ) D E 2 w + ϑ D E 2 b δ + ζ δ ′ ′ ( b δ − w ) D E ( b δ − w ) ⊗ D E ( b δ − w ) ≥ D E 2 w + 2 ϑ I . D_E^2\widetilde w=(1-\vartheta)D_E^2w+\vartheta D_E^2b_\delta+\zeta_\delta''(b_\delta-w)D_E(b_\delta-w)\otimes D_E(b_\delta-w)
\ge D_E^2w+2\vartheta I. D E 2 w = ( 1 − ϑ ) D E 2 w + ϑ D E 2 b δ + ζ δ ′′ ( b δ − w ) D E ( b δ − w ) ⊗ D E ( b δ − w ) ≥ D E 2 w + 2 ϑ I . The Klein formulas express the symmetric Weingarten matrix of a graph v v v A v = T v D E 2 v T v A_v=T_vD_E^2vT_v A v = T v D E 2 v T v T v T_v T v ( x , v , D E v ) (x,v,D_Ev) ( x , v , D E v ) I I I ( x , v , D E v ) = ( 0 , 0 , 0 ) (x,v,D_Ev)=(0,0,0) ( x , v , D E v ) = ( 0 , 0 , 0 )
∥ T w ~ − T w ∥ ≤ C ϑ δ , T w ~ 2 ≥ ( 1 − C δ ) I . \|T_{\widetilde w}-T_w\|\le C\vartheta\sqrt{\delta},\qquad T_{\widetilde w}^2\ge(1-C\sqrt{\delta})I. ∥ T w − T w ∥ ≤ C ϑ δ , T w 2 ≥ ( 1 − C δ ) I . Together with the uniform bound for D E 2 w D_E^2w D E 2 w
A ~ = T w ~ D E 2 w ~ T w ~ ≥ A + ( 2 − C δ ) ϑ I ≥ A + ϑ I (95) \widetilde A=T_{\widetilde w}D_E^2\widetilde wT_{\widetilde w}\ge A+(2-C\sqrt{\delta})\vartheta I\ge A+\vartheta I \tag*{(95)} A = T w D E 2 w T w ≥ A + ( 2 − C δ ) ϑ I ≥ A + ϑ I (95) for sufficiently small δ \delta δ A ~ = A w ~ \widetilde A=A_{\widetilde w} A = A w A = A w A=A_w A = A w
κ ~ i ≥ κ i + ϑ , κ ~ 1 κ ~ 2 ≥ ( κ 1 + ϑ ) ( κ 2 + ϑ ) ≥ ( 1 + ϑ ) 2 , \widetilde\kappa_i\ge\kappa_i+\vartheta,\qquad\widetilde\kappa_1\widetilde\kappa_2\ge(\kappa_1+\vartheta)(\kappa_2+\vartheta)\ge(1+\vartheta)^2, κ i ≥ κ i + ϑ , κ 1 κ 2 ≥ ( κ 1 + ϑ ) ( κ 2 + ϑ ) ≥ ( 1 + ϑ ) 2 , where we used κ 1 κ 2 ≥ 1 \kappa_1\kappa_2 \ge1 κ 1 κ 2 ≥ 1 κ 1 + κ 2 ≥ 2 \kappa_1+\kappa_2 \ge2 κ 1 + κ 2 ≥ 2 ϑ > 0 \vartheta> 0 ϑ > 0 w w w b δ b_\delta b δ o o o
Replacing the lower spherical cap over B ϱ ( 0 ) B_\varrho(0) B ϱ ( 0 ) w ~ \widetilde{w} w Σ \Sigma Σ b δ + ( x ) = 1 4 − δ + 1 16 − ∣ x ∣ 2 b_\delta^+(x)=\frac{1}{4}-\delta+\sqrt{\frac{1}{16}-|x|^2} b δ + ( x ) = 4 1 − δ + 16 1 − ∣ x ∣ 2
0 ≤ w ~ ≤ max { w , b δ } + δ 2 ≤ 4 ϱ 2 + δ 2 < 1 4 < b δ + ( x ) on B ϱ ( 0 ) . 0 \le\widetilde{w} \le\max\{w,b_\delta\}+\delta^2 \le4\varrho^2+\delta^2 < \frac{1}{4} < b_\delta^+(x) \quad\text{on } B_\varrho(0). 0 ≤ w ≤ max { w , b δ } + δ 2 ≤ 4 ϱ 2 + δ 2 < 4 1 < b δ + ( x ) on B ϱ ( 0 ) . Here we used 0 ≤ w ≤ ∣ x ∣ 2 0 \le w \le|x|^2 0 ≤ w ≤ ∣ x ∣ 2 b δ ≤ 4 ∣ x ∣ 2 b_\delta\le4|x|^2 b δ ≤ 4∣ x ∣ 2 B ϱ ( 0 ) B_\varrho(0) B ϱ ( 0 ) δ \delta δ Σ ⊂ B 1 / 2 ( 0 ) \Sigma\subset B_{1/2}(0) Σ ⊂ B 1/2 ( 0 ) M ^ \widehat{M} M
κ i ( M ^ ) > 0 , sec M ^ ≥ 0 , sec M ^ > 0 on M ^ ∖ { o } . (96) \kappa_i(\widehat{M}) > 0,\qquad\sec_{\widehat{M}} \ge0,\qquad\sec_{\widehat{M}} > 0\quad\text{on }\widehat{M}\setminus\{o\}. \tag*{(96)} κ i ( M ) > 0 , sec M ≥ 0 , sec M > 0 on M ∖ { o } . (96) Since M ^ \widehat{M} M o o o (87) is independent of ϕ ( 0 ) \phi(0) ϕ ( 0 ) o o o (92) .
Step 3. Strict approximation and loss of sectional curvature. For the hypersurface Σ \Sigma Σ
M c = { ( 1 , c Y ) 1 − c 2 ∣ Y ∣ 2 ∣ Y ∈ Σ } , 0 < c ≤ 1. (97) M_c=\left\{\left.\frac{(1,cY)}{\sqrt{1-c^2|Y|^2}}\right|Y\in\Sigma\right\},\qquad0<c\le1. \tag*{(97)} M c = { 1 − c 2 ∣ Y ∣ 2 ( 1 , c Y ) Y ∈ Σ } , 0 < c ≤ 1. (97) Then M 1 = M ^ M_1=\widehat{M} M 1 = M
sec M c ≥ c − 2 − 1 > 0 ( 0 < c < 1 ) , M c ⟶ M ^ smoothly as c ↑ 1. (98) \sec_{M_c}\ge c^{-2}-1>0\quad(0<c<1),\qquad M_c\longrightarrow\widehat{M}\quad\text{smoothly as }c\uparrow1. \tag*{(98)} sec M c ≥ c − 2 − 1 > 0 ( 0 < c < 1 ) , M c ⟶ M smoothly as c ↑ 1. (98) The principal curvatures of M ^ \widehat{M} M Γ + \Gamma_+ Γ + [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ] c ≤ 1 c\le1 c ≤ 1 C 2 C^2 C 2
Parametrize these solutions over M ^ \widehat{M} M X c ( p , t ) X_c(p,t) X c ( p , t ) X 1 ( p , 0 ) = p X_1(p,0)=p X 1 ( p , 0 ) = p X c ( p , t ) X_c(p,t) X c ( p , t ) κ 1 , c ( p , t ) , … , κ n , c ( p , t ) \kappa_{1,c}(p,t),\ldots,\kappa_{n,c}(p,t) κ 1 , c ( p , t ) , … , κ n , c ( p , t )
S c ( p , t ) = κ 1 , c ( p , t ) κ 2 , c ( p , t ) − 1. S_c(p,t)=\kappa_{1,c}(p,t)\kappa_{2,c}(p,t)-1. S c ( p , t ) = κ 1 , c ( p , t ) κ 2 , c ( p , t ) − 1. Since ∇ S 1 ( o , 0 ) = 0 \nabla S_1(o,0)=0 ∇ S 1 ( o , 0 ) = 0 (92) and the agreement of M ^ \widehat{M} M o o o
S 1 ( o , 0 ) = 0 , ∂ t S 1 ( o , 0 ) < − 2 ( k + ℓ − n ) n ( n − 1 ) . S_1(o,0)=0,\qquad\partial_tS_1(o,0)<-\frac{2(k+\ell-n)}{n(n-1)}. S 1 ( o , 0 ) = 0 , ∂ t S 1 ( o , 0 ) < − n ( n − 1 ) 2 ( k + ℓ − n ) . The first two principal curvatures are simple at ( o , 0 ) (o,0) ( o , 0 ) T 0 T_0 T 0
S 1 ( o , t ) ≤ − k + ℓ − n n ( n − 1 ) t , 0 ≤ t ≤ T 0 . S_1(o,t)\le-\frac{k+\ell-n}{n(n-1)}t,\qquad0\le t\le T_0. S 1 ( o , t ) ≤ − n ( n − 1 ) k + ℓ − n t , 0 ≤ t ≤ T 0 . Given T > 0 T>0 T > 0 0 < t ∗ < min { T , T 0 } 0<t_*<\min\{T,T_0\} 0 < t ∗ < min { T , T 0 } C 2 C^2 C 2
S c ( o , t ∗ ) ⟶ S 1 ( o , t ∗ ) as c ↑ 1. S_c(o,t_*)\longrightarrow S_1(o,t_*)\quad\text{as }c\uparrow1. S c ( o , t ∗ ) ⟶ S 1 ( o , t ∗ ) as c ↑ 1. Thus, for c < 1 c<1 c < 1
S c ( o , t ∗ ) < − k + ℓ − n 2 n ( n − 1 ) t ∗ < 0. S_c(o,t_*)<-\frac{k+\ell-n}{2n(n-1)}t_*<0. S c ( o , t ∗ ) < − 2 n ( n − 1 ) k + ℓ − n t ∗ < 0. Together with (98) , this gives a smooth closed initial hypersurface with positive sectional curvature whose solution develops negative sectional curvature before time T T T T > 0 T>0 T > 0
A paired concavity inequality
We prove the estimate used in Lemma 3.3 . We first reduce the quadratic form to a single derivative estimate. The remaining subsections establish that estimate using elementary symmetric polynomials. Throughout this section, q i = ∂ i q q_i=\partial_iq q i = ∂ i q q i j = ∂ i ∂ j q q_{ij}=\partial_i\partial_jq q ij = ∂ i ∂ j q
Proposition 6.1. Let n ≥ 3 n\ge3 n ≥ 3 1 ≤ s < r ≤ n 1\le s<r\le n 1 ≤ s < r ≤ n
q = ( σ r σ s ) 1 / ( r − s ) . q=\left(\frac{\sigma_r}{\sigma_s}\right)^{1/(r-s)}. q = ( σ s σ r ) 1/ ( r − s ) . The inequality
J [ q ] ( λ , ξ ) = − D 2 q ( λ ) [ ξ , ξ ] + 2 ∑ i = 2 n q i ( λ ) λ 1 − λ i ξ i 2 − 2 q 1 ( λ ) λ 1 − λ 2 ξ 1 2 ≥ 0 (99) \mathcal{J}[q](\lambda,\xi)=-D^2q(\lambda)[\xi,\xi]+2\sum_{i=2}^{n}\frac{q_i(\lambda)}{\lambda_1-\lambda_i}\xi_i^2-\frac{2q_1(\lambda)}{\lambda_1-\lambda_2}\xi_1^2\ge0 \tag*{(99)} J [ q ] ( λ , ξ ) = − D 2 q ( λ ) [ ξ , ξ ] + 2 i = 2 ∑ n λ 1 − λ i q i ( λ ) ξ i 2 − λ 1 − λ 2 2 q 1 ( λ ) ξ 1 2 ≥ 0 (99) holds for every
λ 1 > λ 2 ≥ ⋯ ≥ λ n > 0 \lambda_1>\lambda_2\ge\cdots\ge\lambda_n>0 λ 1 > λ 2 ≥ ⋯ ≥ λ n > 0 and ξ ∈ R n \xi\in\mathbb{R}^n ξ ∈ R n
λ 2 ξ 1 + λ 1 ξ 2 = 0 , (100) \lambda_2\xi_1+\lambda_1\xi_2=0, \tag*{(100)} λ 2 ξ 1 + λ 1 ξ 2 = 0 , (100) if and only if r + s ≥ n r+s\ge n r + s ≥ n ξ = 0 \xi=0 ξ = 0
The adjacent quotients are concave on Γ + \Gamma_+ Γ + [20 Theorem 2.5] . Their geometric mean
q = ( ∏ j = s + 1 r σ j σ j − 1 ) 1 / ( r − s ) (101) q=\left(\prod_{j=s+1}^{r}\frac{\sigma_j}{\sigma_{j-1}}\right)^{1/(r-s)} \tag*{(101)} q = ( j = s + 1 ∏ r σ j − 1 σ j ) 1/ ( r − s ) (101) is therefore concave, and Newton’s inequalities give q i > 0 q_i>0 q i > 0 [23 Theorem 1.1] , [10 Lemma 1.1] , and [33 Lemma 1.1] .
Reduction of the quadratic form
By homogeneity, we may normalize
λ = ( a , 1 , z ) , a > 1 , z = ( z 3 , … , z n ) ∈ ( 0 , 1 ] n − 2 . \lambda=(a,1,z),\qquad a>1,\qquad z=(z_3,\ldots,z_n)\in(0,1]^{n-2}. λ = ( a , 1 , z ) , a > 1 , z = ( z 3 , … , z n ) ∈ ( 0 , 1 ] n − 2 . The constraint becomes ξ 2 = − ξ 1 / a \xi_2=-\xi_1/a ξ 2 = − ξ 1 / a
J [ q ] = − D 2 q [ ξ , ξ ] + ∑ i = 3 n 2 q i a − z i ξ i 2 − 2 a − 1 ( q 1 − q 2 a 2 ) ξ 1 2 . (102) \mathcal{J}[q]=-D^2q[\xi,\xi]+\sum_{i=3}^{n}\frac{2q_i}{a-z_i}\xi_i^2-\frac{2}{a-1}\left(q_1-\frac{q_2}{a^2}\right)\xi_1^2. \tag*{(102)} J [ q ] = − D 2 q [ ξ , ξ ] + i = 3 ∑ n a − z i 2 q i ξ i 2 − a − 1 2 ( q 1 − a 2 q 2 ) ξ 1 2 . (102) The first two terms are nonnegative.
Proof of Proposition 6.1. For sufficiency, we assume first that r + s ≥ n r+s\ge n r + s ≥ n
v = ( a , − a q 1 q 2 , 0 , … , 0 ) , c i = − D 2 q [ v , e i ] , 1 ≤ i ≤ n , v=\left(a,-a\frac{q_1}{q_2},0,\ldots,0\right),\qquad c_i=-D^2q[v,e_i],\qquad1\le i\le n, v = ( a , − a q 2 q 1 , 0 , … , 0 ) , c i = − D 2 q [ v , e i ] , 1 ≤ i ≤ n , where e i e_i e i i i i (100) gives
( c 1 − c 2 a ) ξ 1 = − D 2 q [ v , ξ ] − ∑ i = 3 n c i ξ i . (103) \left(c_1-\frac{c_2}{a}\right)\xi_1=-D^2q[v,\xi]-\sum_{i=3}^{n}c_i\xi_i. \tag*{(103)} ( c 1 − a c 2 ) ξ 1 = − D 2 q [ v , ξ ] − i = 3 ∑ n c i ξ i . (103) For any real τ \tau τ ξ = τ v + ( ξ − τ v ) \xi=\tau v+(\xi-\tau v) ξ = τ v + ( ξ − τ v ) (103) , we have
− D 2 q [ ξ , ξ ] + ∑ i = 3 n 2 q i a − z i ξ i 2 = − D 2 q [ ξ − τ v , ξ − τ v ] + ∑ i = 3 n 2 q i a − z i ( ξ i + τ a − z i 2 q i c i ) 2 − 2 τ D 2 q [ v , ξ ] − 2 τ ∑ i = 3 n c i ξ i − τ 2 ( − D 2 q [ v , v ] + ∑ i = 3 n a − z i 2 q i c i 2 ) \begin{aligned}
-D^2q[\xi,\xi]+\sum_{i=3}^{n}\frac{2q_i}{a-z_i}\xi_i^2 \\
&=-D^2q[\xi-\tau v,\xi-\tau v]+\sum_{i=3}^{n}\frac{2q_i}{a-z_i}\left(\xi_i+\tau\frac{a-z_i}{2q_i}c_i\right)^2
-2\tau D^2q[v,\xi]-2\tau\sum_{i=3}^{n}c_i\xi_i-\tau^2\left(-D^2q[v,v]+\sum_{i=3}^{n}\frac{a-z_i}{2q_i}c_i^2\right)
\end{aligned} − D 2 q [ ξ , ξ ] + i = 3 ∑ n a − z i 2 q i ξ i 2 = − D 2 q [ ξ − τ v , ξ − τ v ] + i = 3 ∑ n a − z i 2 q i ( ξ i + τ 2 q i a − z i c i ) 2 − 2 τ D 2 q [ v , ξ ] − 2 τ i = 3 ∑ n c i ξ i − τ 2 ( − D 2 q [ v , v ] + i = 3 ∑ n 2 q i a − z i c i 2 ) ≥ 2 τ ( c 1 − c 2 a ) ξ 1 − τ 2 ( − D 2 q [ v , v ] + ∑ i = 3 n a − z i 2 q i c i 2 ) . \geq2\tau\left(c_1-\frac{c_2}{a}\right)\xi_1-\tau^2\left(-D^2q[v,v]+\sum_{i=3}^{n}\frac{a-z_i}{2q_i}c_i^2\right). ≥ 2 τ ( c 1 − a c 2 ) ξ 1 − τ 2 ( − D 2 q [ v , v ] + i = 3 ∑ n 2 q i a − z i c i 2 ) . where we used the concavity of q q q τ \tau τ
− D 2 q [ ξ , ξ ] + ∑ i = 3 n 2 q i a − z i ξ i 2 ≥ ( c 1 − c 2 a ) 2 ξ 1 2 − D 2 q [ v , v ] + ∑ i = 3 n a − z i 2 q i c i 2 (104) -D^2q[\xi,\xi]+\sum_{i=3}^{n}\frac{2q_i}{a-z_i}\xi_i^2\geq\frac{\left(c_1-\frac{c_2}{a}\right)^2\xi_1^2}{-D^2q[v,v]+\sum_{i=3}^{n}\frac{a-z_i}{2q_i}c_i^2} \tag*{(104)} − D 2 q [ ξ , ξ ] + i = 3 ∑ n a − z i 2 q i ξ i 2 ≥ − D 2 q [ v , v ] + ∑ i = 3 n 2 q i a − z i c i 2 ( c 1 − a c 2 ) 2 ξ 1 2 (104) To control the last term of (102) , in view of (104) , we establish the following lemma in Subsection 6.3.
Lemma 6.2. Suppose r + s ≥ n r+s\geq n r + s ≥ n λ = ( a , 1 , z ) \lambda=(a,1,z) λ = ( a , 1 , z ) − D 2 q [ v , v ] > 0 -D^2q[v,v]>0 − D 2 q [ v , v ] > 0
( c 1 − c 2 a ) 2 > 2 a − 1 ( q 1 − q 2 a 2 ) [ − D 2 q [ v , v ] + ∑ i = 3 n a − z i 2 q i c i 2 ] . (105) \left(c_1-\frac{c_2}{a}\right)^2>\frac{2}{a-1}\left(q_1-\frac{q_2}{a^2}\right)\left[-D^2q[v,v]+\sum_{i=3}^{n}\frac{a-z_i}{2q_i}c_i^2\right]. \tag*{(105)} ( c 1 − a c 2 ) 2 > a − 1 2 ( q 1 − a 2 q 2 ) [ − D 2 q [ v , v ] + i = 3 ∑ n 2 q i a − z i c i 2 ] . (105) If ξ 1 = 0 \xi_1=0 ξ 1 = 0 (100) gives ξ 2 = 0 \xi_2=0 ξ 2 = 0 (102) is strictly positive for every nonzero ξ \xi ξ q i > 0 q_i>0 q i > 0 J [ q ] ( λ , ξ ) ≥ 0 \mathcal{J}[q](\lambda,\xi)\geq0 J [ q ] ( λ , ξ ) ≥ 0 ξ = 0 \xi=0 ξ = 0
For necessity, suppose that r + s < n r+s<n r + s < n (99) holds for all admissible ( λ , ξ ) (\lambda,\xi) ( λ , ξ ) 1 = ( 1 , … , 1 ) \mathbf{1}=(1,\ldots,1) 1 = ( 1 , … , 1 )
q i = q ( 1 ) n , q i i = − q ( 1 ) ( r + s − 1 ) n 2 , q i j = q ( 1 ) ( r + s − 1 ) n 2 ( n − 1 ) ( i ≠ j ) . (106) q_i=\frac{q(1)}{n},\qquad q_{ii}=-\frac{q(1)(r+s-1)}{n^2},\qquad q_{ij}=\frac{q(1)(r+s-1)}{n^2(n-1)}\quad(i\ne j). \tag*{(106)} q i = n q ( 1 ) , q ii = − n 2 q ( 1 ) ( r + s − 1 ) , q ij = n 2 ( n − 1 ) q ( 1 ) ( r + s − 1 ) ( i = j ) . (106) Taking λ ( a ) = ( a , 1 , … , 1 ) \lambda(a)=(a,1,\ldots,1) λ ( a ) = ( a , 1 , … , 1 ) ξ ( a ) = ( a , − 1 , 0 , … , 0 ) \xi(a)=(a,-1,0,\ldots,0) ξ ( a ) = ( a , − 1 , 0 , … , 0 )
J [ q ] ( λ ( a ) , ξ ( a ) ) = − D 2 q [ ξ ( a ) , ξ ( a ) ] − 2 a 2 q 1 ( λ ( a ) ) − q 2 ( λ ( a ) ) a − 1 . \mathcal{J}[q](\lambda(a),\xi(a))=-D^2q[\xi(a),\xi(a)]-2\frac{a^2q_1(\lambda(a))-q_2(\lambda(a))}{a-1}. J [ q ] ( λ ( a ) , ξ ( a )) = − D 2 q [ ξ ( a ) , ξ ( a )] − 2 a − 1 a 2 q 1 ( λ ( a )) − q 2 ( λ ( a )) . By (106) , the two terms tend to 2 q ( 1 ) ( r + s − 1 ) / ( n ( n − 1 ) ) 2q(1)(r+s-1)/(n(n-1)) 2 q ( 1 ) ( r + s − 1 ) / ( n ( n − 1 )) − 2 q ( 1 ) ( 2 n − r − s − 1 ) / ( n ( n − 1 ) ) -2q(1)(2n-r-s-1)/(n(n-1)) − 2 q ( 1 ) ( 2 n − r − s − 1 ) / ( n ( n − 1 ))
lim a ↓ 1 J [ q ] ( λ ( a ) , ξ ( a ) ) = 4 q ( 1 ) ( r + s − n ) n ( n − 1 ) < 0. (107) \lim_{a\downarrow1}\mathcal{J}[q](\lambda(a),\xi(a))=\frac{4q(1)(r+s-n)}{n(n-1)}<0. \tag*{(107)} a ↓ 1 lim J [ q ] ( λ ( a ) , ξ ( a )) = n ( n − 1 ) 4 q ( 1 ) ( r + s − n ) < 0. (107) Hence J [ q ] ( λ ( a ) , ξ ( a ) ) < 0 \mathcal{J}[q](\lambda(a),\xi(a))<0 J [ q ] ( λ ( a ) , ξ ( a )) < 0 a > 1 a>1 a > 1 1 1 1
Symmetric polynomial estimates
For a positive vector z z z N N N σ j ( z ) \sigma_j(z) σ j ( z ) σ 0 = 1 \sigma_0=1 σ 0 = 1 σ j = 0 \sigma_j=0 σ j = 0 0 ≤ j ≤ N 0\leq j\leq N 0 ≤ j ≤ N 1 ≤ s < r ≤ N + 2 1\leq s<r\leq N+2 1 ≤ s < r ≤ N + 2
P = σ r − 2 σ s − 1 − σ r − 1 σ s − 2 , P=\sigma_{r-2}\sigma_{s-1}-\sigma_{r-1}\sigma_{s-2}, P = σ r − 2 σ s − 1 − σ r − 1 σ s − 2 , Q = σ r − 2 σ s − σ r σ s − 2 , R = σ r − 1 σ s − σ r σ s − 1 , Q=\sigma_{r-2}\sigma_s-\sigma_r\sigma_{s-2},\qquad R=\sigma_{r-1}\sigma_s-\sigma_r\sigma_{s-1}, Q = σ r − 2 σ s − σ r σ s − 2 , R = σ r − 1 σ s − σ r σ s − 1 , t = Q / P , u = R / P . (108) t=Q/P,\qquad u=R/P. \tag*{(108)} t = Q / P , u = R / P . (108) All symmetric polynomials in these expressions are evaluated at z z z P > 0 P>0 P > 0 Q , R ≥ 0 Q,R\geq0 Q , R ≥ 0 t i = ∂ i t t_i=\partial_i t t i = ∂ i t u i = ∂ i u u_i=\partial_i u u i = ∂ i u z ∣ i z|i z ∣ i z i z_i z i
Lemma 6.3. For every positive z z z
0 ≤ t i ≤ 1 , 0 ≤ u i ≤ { t , r − s ≥ 2 , 2 t , r − s = 1. (109) 0 \le t_i \le1,\qquad0 \le u_i \le
\begin{cases}
t, & r-s \ge2,\\
2t, & r-s=1.
\end{cases} \tag*{(109)} 0 ≤ t i ≤ 1 , 0 ≤ u i ≤ { t , 2 t , r − s ≥ 2 , r − s = 1. (109) In particular, if N = n − 2 N=n-2 N = n − 2 r + s ≥ n r+s\ge n r + s ≥ n z ∈ ( 0 , 1 ] N z\in(0,1]^N z ∈ ( 0 , 1 ] N
0 ≤ u ≤ ( n − r ) ( n − s ) r s ≤ 1 , (110) 0 \le u \le\frac{(n-r)(n-s)}{rs} \le1, \tag*{(110)} 0 ≤ u ≤ r s ( n − r ) ( n − s ) ≤ 1 , (110) and if r = s + 1 r=s+1 r = s + 1 0 ≤ t ≤ 2 ( n − s − 1 ) s + 1 < 2 0\le t\le\frac{2(n-s-1)}{s+1}<2 0 ≤ t ≤ s + 1 2 ( n − s − 1 ) < 2
Proof. We use the minors
Δ p , q = σ p − 1 σ q − σ p σ q − 1 , Δ p , q , j = det ( σ p σ p − 1 σ p − 2 σ q σ q − 1 σ q − 2 σ j σ j − 1 σ j − 2 ) . \Delta_{p,q}=\sigma_{p-1}\sigma_q-\sigma_p\sigma_{q-1},\qquad
\Delta_{p,q,j}=\det\begin{pmatrix}
\sigma_p & \sigma_{p-1} & \sigma_{p-2}\\
\sigma_q & \sigma_{q-1} & \sigma_{q-2}\\
\sigma_j & \sigma_{j-1} & \sigma_{j-2}
\end{pmatrix}. Δ p , q = σ p − 1 σ q − σ p σ q − 1 , Δ p , q , j = det σ p σ q σ j σ p − 1 σ q − 1 σ j − 1 σ p − 2 σ q − 2 σ j − 2 . The second minor is nonnegative for 0 ≤ p ≤ q ≤ j 0\le p\le q\le j 0 ≤ p ≤ q ≤ j ( σ p − q ) 0 ≤ p , q ≤ L (\sigma_{p-q})_{0\le p,q\le L} ( σ p − q ) 0 ≤ p , q ≤ L L ≥ max { j , 2 } L\ge\max\{j,2\} L ≥ max { j , 2 }
( σ p − q ) p , q = ∏ h ( δ p q + z h δ p , q + 1 ) p , q . (\sigma_{p-q})_{p,q}=\prod_h(\delta_{pq}+z_h\delta_{p,q+1})_{p,q}. ( σ p − q ) p , q = h ∏ ( δ pq + z h δ p , q + 1 ) p , q . Each bidiagonal factor has nonnegative minors, since a nonzero permutation term must preserve the row and column order. Cauchy–Binet gives the assertion for the product.
Fix i i i τ = z i \tau=z_i τ = z i x = z ∣ i x=z|i x = z ∣ i σ j ( z ) = σ j ( x ) + τ σ j − 1 ( x ) \sigma_j(z)=\sigma_j(x)+\tau\sigma_{j-1}(x) σ j ( z ) = σ j ( x ) + τ σ j − 1 ( x )
P ( τ ) = α + β τ + γ τ 2 , P(\tau)=\alpha+\beta\tau+\gamma\tau^2, P ( τ ) = α + β τ + γ τ 2 , Q ( τ ) = η + θ τ + β τ 2 , R ( τ ) = δ + η τ + α τ 2 , Q(\tau)=\eta+\theta\tau+\beta\tau^2,\qquad R(\tau)=\delta+\eta\tau+\alpha\tau^2, Q ( τ ) = η + θ τ + β τ 2 , R ( τ ) = δ + η τ + α τ 2 , α = Δ r − 1 , s − 1 , γ = Δ r − 2 , s − 2 , δ = Δ r , s , \alpha=\Delta_{r-1,s-1},\qquad\gamma=\Delta_{r-2,s-2},\qquad\delta=\Delta_{r,s}, α = Δ r − 1 , s − 1 , γ = Δ r − 2 , s − 2 , δ = Δ r , s , β = Δ r − 2 , s − 1 + Δ r − 1 , s − 2 , η = Δ r − 1 , s + Δ r , s − 1 , \beta=\Delta_{r-2,s-1}+\Delta_{r-1,s-2},\qquad
\eta=\Delta_{r-1,s}+\Delta_{r,s-1}, β = Δ r − 2 , s − 1 + Δ r − 1 , s − 2 , η = Δ r − 1 , s + Δ r , s − 1 , θ = 2 α + Δ r − 2 , s + Δ r , s − 2 . (111) \theta=2\alpha+\Delta_{r-2,s}+\Delta_{r,s-2}. \tag*{(111)} θ = 2 α + Δ r − 2 , s + Δ r , s − 2 . (111) Set Δ 1 = Δ s − 1 , s , r − 1 \Delta_1=\Delta_{s-1,s,r-1} Δ 1 = Δ s − 1 , s , r − 1 Δ 2 = Δ s − 1 , r − 1 , r \Delta_2=\Delta_{s-1,r-1,r} Δ 2 = Δ s − 1 , r − 1 , r
α θ − β η = α 2 − γ δ = σ s − 1 Δ 2 + σ r − 1 Δ 1 ≥ 0 , \alpha\theta-\beta\eta=\alpha^2-\gamma\delta
=\sigma_{s-1}\Delta_2+\sigma_{r-1}\Delta_1\ge0, α θ − β η = α 2 − γ δ = σ s − 1 Δ 2 + σ r − 1 Δ 1 ≥ 0 , α β − γ η = σ r − 2 Δ 1 + σ s − 2 Δ 2 ≥ 0 , \alpha\beta-\gamma\eta=\sigma_{r-2}\Delta_1+\sigma_{s-2}\Delta_2\ge0, α β − γ η = σ r − 2 Δ 1 + σ s − 2 Δ 2 ≥ 0 , β 2 − γ θ = σ r − 3 Δ 1 + σ s − 3 Δ 2 ≥ 0 , \beta^2-\gamma\theta=\sigma_{r-3}\Delta_1+\sigma_{s-3}\Delta_2\ge0, β 2 − γ θ = σ r − 3 Δ 1 + σ s − 3 Δ 2 ≥ 0 , α η − β δ = σ s − 1 Δ s , r − 1 , r + σ r − 1 Δ s − 1 , s , r ≥ 0. (112) \alpha\eta-\beta\delta
=\sigma_{s-1}\Delta_{s,r-1,r}+\sigma_{r-1}\Delta_{s-1,s,r}\ge0. \tag*{(112)} α η − β δ = σ s − 1 Δ s , r − 1 , r + σ r − 1 Δ s − 1 , s , r ≥ 0. (112) Primes denote derivatives in τ \tau τ
P Q ′ − Q P ′ = ( α 2 − γ δ ) + 2 ( α β − γ η ) τ + ( β 2 − γ θ ) τ 2 ≥ 0 , PQ'-QP'=(\alpha^2-\gamma\delta)+2(\alpha\beta-\gamma\eta)\tau+(\beta^2-\gamma\theta)\tau^2\ge0, P Q ′ − Q P ′ = ( α 2 − γ δ ) + 2 ( α β − γ η ) τ + ( β 2 − γ θ ) τ 2 ≥ 0 , P R ′ − R P ′ = ( α η − β δ ) + 2 ( α 2 − γ δ ) τ + ( α β − γ η ) τ 2 ≥ 0. (113) PR'-RP'=(\alpha\eta-\beta\delta)+2(\alpha^2-\gamma\delta)\tau+(\alpha\beta-\gamma\eta)\tau^2\ge0. \tag*{(113)} P R ′ − R P ′ = ( α η − β δ ) + 2 ( α 2 − γ δ ) τ + ( α β − γ η ) τ 2 ≥ 0. (113) These prove t i , u i ≥ 0 t_i,u_i\ge0 t i , u i ≥ 0
P 2 ( 1 − t i ) = γ ( R + τ Q + τ 2 P ) ≥ 0 P^2(1-t_i)=\gamma(R+\tau Q+\tau^2P)\ge0 P 2 ( 1 − t i ) = γ ( R + τ Q + τ 2 P ) ≥ 0 and
P 2 ( t − u i ) = P [ ( θ − 2 α ) τ + β τ 2 ] + R ( β + 2 γ τ ) ≥ 0 ( r − s ≥ 2 ) , P^2(t-u_i)=P[(\theta-2\alpha)\tau+\beta\tau^2]+R(\beta+2\gamma\tau)\ge0
\qquad(r-s\ge2), P 2 ( t − u i ) = P [( θ − 2 α ) τ + β τ 2 ] + R ( β + 2 γ τ ) ≥ 0 ( r − s ≥ 2 ) , P 2 ( 2 t − u i ) = P [ η + 2 ( θ − α ) τ + 2 β τ 2 ] + R ( β + 2 γ τ ) ≥ 0 ( r − s = 1 ) . P^2(2t-u_i)=P[\eta+2(\theta-\alpha)\tau+2\beta\tau^2]+R(\beta+2\gamma\tau)\ge0
\qquad(r-s=1). P 2 ( 2 t − u i ) = P [ η + 2 ( θ − α ) τ + 2 β τ 2 ] + R ( β + 2 γ τ ) ≥ 0 ( r − s = 1 ) . Here θ ≥ 2 α \theta\ge2\alpha θ ≥ 2 α θ ≥ α \theta\ge\alpha θ ≥ α z = 1 z=1 z = 1 (110) . □
Lemma 6.4. Suppose N = n − 2 N=n-2 N = n − 2 r + s ≥ n r+s\ge n r + s ≥ n z ∈ ( 0 , 1 ] N \mathbf{z}\in(0,1]^N z ∈ ( 0 , 1 ] N z ( i ) \mathbf{z}^{(i)} z ( i ) z \mathbf{z} z z i z_i z i 1 1 1
ω i = P ( z ) P ( z ( i ) ) . (114) \omega_i=\frac{P(\mathbf{z})}{P(\mathbf{z}^{(i)})}. \tag*{(114)} ω i = P ( z ( i ) ) P ( z ) . (114) Then
∑ i ω i t i ≤ t + 2 , ∑ i ω i u i ≤ 4. (115) \sum_i\omega_i t_i\le t+2,\qquad\sum_i\omega_i u_i\le4. \tag*{(115)} i ∑ ω i t i ≤ t + 2 , i ∑ ω i u i ≤ 4. (115) Proof. We first assume n = r + s n=r+s n = r + s P P P J J J n − 3 − 2 J n-3-2J n − 3 − 2 J J + 1 J+1 J + 1
p J = ( n − 3 − 2 J r − 2 − J ) − ( n − 3 − 2 J r − 1 − J ) = ( r − s ) ( n − 3 − 2 J ) ! ( r − 1 − J ) ! ( s − 1 − J ) ! > 0 , 0 ≤ J ≤ s − 1. (116) p_J=\binom{n-3-2J}{r-2-J}-\binom{n-3-2J}{r-1-J}=\frac{(r-s)(n-3-2J)!}{(r-1-J)!(s-1-J)!}>0,\qquad0\le J\le s-1. \tag*{(116)} p J = ( r − 2 − J n − 3 − 2 J ) − ( r − 1 − J n − 3 − 2 J ) = ( r − 1 − J )! ( s − 1 − J )! ( r − s ) ( n − 3 − 2 J )! > 0 , 0 ≤ J ≤ s − 1. (116) Distribute each monomial equally among its zero coordinates,
C i = ∑ m i = 0 p J z m J + 1 > 0 , ∑ i C i = P . (117) C_i=\sum_{m_i=0}\frac{p_J\mathbf{z}^{\mathbf{m}}}{J+1}>0,\qquad\sum_i C_i=P. \tag*{(117)} C i = m i = 0 ∑ J + 1 p J z m > 0 , i ∑ C i = P . (117) Here J J J m \mathbf{m} m (116) count assignments to the first factor of each product defining P P P t t t
( ω i − z i ) t i ≤ 2 C i P , ω i u i ≤ 4 C i P . (\omega_i-z_i)t_i\le\frac{2C_i}{P},\qquad\omega_i u_i\le\frac{4C_i}{P}. ( ω i − z i ) t i ≤ P 2 C i , ω i u i ≤ P 4 C i . Indeed, summation uses ∑ i z i t i = t \sum_i z_i t_i=t ∑ i z i t i = t ∑ i C i = P \sum_i C_i=P ∑ i C i = P
Fix i i i x = z ∣ i x=z|i x = z ∣ i g = C i g=C_i g = C i (111) . On x x x J J J J J J n − 3 − 2 J n-3-2J n − 3 − 2 J
g = ∑ m p J x m J + 1 , α = ∑ m p J x m , g=\sum_{\mathbf{m}}\frac{p_Jx^{\mathbf{m}}}{J+1},\qquad\alpha=\sum_{\mathbf{m}}p_Jx^{\mathbf{m}}, g = m ∑ J + 1 p J x m , α = m ∑ p J x m , γ = ∑ m p J x m J + 1 ∑ m h = 2 x h − 2 , δ = ∑ m p J x m J + 1 ∑ m h = 0 x h 2 , \gamma=\sum_{\mathbf{m}}\frac{p_Jx^{\mathbf{m}}}{J+1}\sum_{m_h=2}x_h^{-2},\qquad\delta=\sum_{\mathbf{m}}\frac{p_Jx^{\mathbf{m}}}{J+1}\sum_{m_h=0}x_h^2, γ = m ∑ J + 1 p J x m m h = 2 ∑ x h − 2 , δ = m ∑ J + 1 p J x m m h = 0 ∑ x h 2 , β = ∑ m p J x m J + 1 ∑ m h = 1 x h − 1 , η = ∑ m p J x m J + 1 ∑ m h = 1 x h . (118) \beta=\sum_{\mathbf{m}}\frac{p_Jx^{\mathbf{m}}}{J+1}\sum_{m_h=1}x_h^{-1},\qquad\eta=\sum_{\mathbf{m}}\frac{p_Jx^{\mathbf{m}}}{J+1}\sum_{m_h=1}x_h. \tag*{(118)} β = m ∑ J + 1 p J x m m h = 1 ∑ x h − 1 , η = m ∑ J + 1 p J x m m h = 1 ∑ x h . (118) For the last four identities, change one exponent by 2 → 0 2\to0 2 → 0 0 → 2 0\to2 0 → 2 1 → 0 1\to0 1 → 0 1 → 2 1\to2 1 → 2 J + 1 J+1 J + 1
δ ≤ α − g , η ≤ β , α ≥ g ≥ α / s > 0. (119) \delta\le\alpha-g,\qquad\eta\le\beta,\qquad\alpha\ge g\ge\alpha/s>0. \tag*{(119)} δ ≤ α − g , η ≤ β , α ≥ g ≥ α / s > 0. (119) We also claim
γ ( δ + g ) ≥ ( α − g ) 2 , α β − γ η ≤ g ( β + 2 γ ) . (120) \gamma(\delta+g)\ge(\alpha-g)^2,\qquad\alpha\beta-\gamma\eta\le g(\beta+2\gamma). \tag*{(120)} γ ( δ + g ) ≥ ( α − g ) 2 , α β − γ η ≤ g ( β + 2 γ ) . (120) For the first estimate, group the monomials by J J J 2 J 2J 2 J y h = x h 2 y_h=x_h^2 y h = x h 2
c σ J ( y ) , c J σ J ( y ) , c ( J + 1 ) σ J − 1 ( y ) , c ( J + 1 ) σ J + 1 ( y ) c\sigma_J(y),\qquad cJ\sigma_J(y),\qquad c(J+1)\sigma_{J-1}(y),\qquad c(J+1)\sigma_{J+1}(y) c σ J ( y ) , c J σ J ( y ) , c ( J + 1 ) σ J − 1 ( y ) , c ( J + 1 ) σ J + 1 ( y ) to g g g α − g \alpha-g α − g γ \gamma γ δ \delta δ c > 0 c>0 c > 0 J ≥ 1 J\ge1 J ≥ 1 σ J / σ J − 1 \sigma_J/\sigma_{J-1} σ J / σ J − 1 0 < y h ≤ 1 0<y_h\le1 0 < y h ≤ 1
∑ h y h 2 ∂ h log σ J σ J − 1 ≤ 1. \sum_h y_h^2\partial_h\log\frac{\sigma_J}{\sigma_{J-1}}\le1. h ∑ y h 2 ∂ h log σ J − 1 σ J ≤ 1. The identity
∑ h y h 2 ∂ h σ j = σ 1 σ j − ( j + 1 ) σ j + 1 (121) \sum_h y_h^2\partial_h\sigma_j=\sigma_1\sigma_j-(j+1)\sigma_{j+1} \tag*{(121)} h ∑ y h 2 ∂ h σ j = σ 1 σ j − ( j + 1 ) σ j + 1 (121) then gives
( J + 1 ) σ J − 1 [ ( J + 1 ) σ J + 1 + σ J ] ≥ J ( J + 1 ) σ J 2 (J+1)\sigma_{J-1}\left[(J+1)\sigma_{J+1}+\sigma_J\right]\ge J(J+1)\sigma_J^2 ( J + 1 ) σ J − 1 [ ( J + 1 ) σ J + 1 + σ J ] ≥ J ( J + 1 ) σ J 2 For J = 0 J=0 J = 0 α − g \alpha-g α − g γ \gamma γ δ \delta δ
For the second estimate, if s = 1 s=1 s = 1 g = α g=\alpha g = α γ = 0 \gamma=0 γ = 0 s ≥ 2 s\ge2 s ≥ 2 γ > 0 \gamma>0 γ > 0 Lemma 6.3 , applied with indices ( r − 1 , s − 1 ) (r-1,s-1) ( r − 1 , s − 1 ) x x x α / γ \alpha/\gamma α / γ
∑ h x h 2 ∂ h log α γ ≤ 2. \sum_h x_h^2\partial_h\log\frac{\alpha}{\gamma}\le2. h ∑ x h 2 ∂ h log γ α ≤ 2. Using (121) and expanding the minors, we obtain the identity
s ( α β − γ η ) = α β + α γ ∑ h x h 2 ∂ h log α γ − ( r − s ) σ s − 2 Δ 2 ≤ α ( β + 2 γ ) ≤ s g ( β + 2 γ ) . \begin{aligned}
s(\alpha\beta-\gamma\eta)=\alpha\beta+\alpha\gamma\sum_h x_h^2\partial_h\log\frac{\alpha}{\gamma}-(r-s)\sigma_{s-2}\Delta_2 \\
&\le\alpha(\beta+2\gamma)\le sg(\beta+2\gamma).
\end{aligned} s ( α β − γ η ) = α β + α γ h ∑ x h 2 ∂ h log γ α − ( r − s ) σ s − 2 Δ 2 ≤ α ( β + 2 γ ) ≤ s g ( β + 2 γ ) . This proves (120) .
We now estimate the derivatives in the fixed coordinate τ = z i \tau=z_i τ = z i
P Q ′ − Q P ′ = ( α 2 − γ δ ) ( 1 − γ τ 2 α ) + ( α β − γ η ) ( 2 τ + β τ 2 α ) , PQ'-QP'=(\alpha^2-\gamma\delta)\left(1-\frac{\gamma\tau^2}{\alpha}\right)+(\alpha\beta-\gamma\eta)\left(2\tau+\frac{\beta\tau^2}{\alpha}\right), P Q ′ − Q P ′ = ( α 2 − γ δ ) ( 1 − α γ τ 2 ) + ( α β − γ η ) ( 2 τ + α β τ 2 ) , ω i − τ = ( 1 − τ ) ( α − γ τ ) P ( 1 ) . \omega_i-\tau=\frac{(1-\tau)(\alpha-\gamma\tau)}{P(1)}. ω i − τ = P ( 1 ) ( 1 − τ ) ( α − γ τ ) . Since P Q ′ − Q P ′ ≥ 0 PQ'-QP'\ge0 P Q ′ − Q P ′ ≥ 0 0 < τ ≤ 1 0<\tau\le1 0 < τ ≤ 1
( 1 − τ ) ( α − γ τ ) ( P Q ′ − Q P ′ ) ≤ α ( 1 − τ ) ( P Q ′ − Q P ′ ) . (1-\tau)(\alpha-\gamma\tau)(PQ'-QP')\le\alpha(1-\tau)(PQ'-QP'). ( 1 − τ ) ( α − γ τ ) ( P Q ′ − Q P ′ ) ≤ α ( 1 − τ ) ( P Q ′ − Q P ′ ) . The two estimates in (120) give
α 2 − γ δ ≤ g ( 2 α + γ − g ) , α β − γ η ≤ g ( β + 2 γ ) . \alpha^2-\gamma\delta\le g(2\alpha+\gamma-g),\qquad\alpha\beta-\gamma\eta\le g(\beta+2\gamma). α 2 − γ δ ≤ g ( 2 α + γ − g ) , α β − γ η ≤ g ( β + 2 γ ) . Using 1 − γ τ 2 / α ≤ 1 1-\gamma\tau^2/\alpha\le1 1 − γ τ 2 / α ≤ 1 P Q ′ − Q P ′ PQ'-QP' P Q ′ − Q P ′
2 g P ( 1 ) P ( τ ) − ( 1 − τ ) ( α − γ τ ) ( P Q ′ − Q P ′ ) ≥ g [ 2 α 2 τ + α g ( 1 − τ ) + α γ ( 1 − 3 τ + 6 τ 2 ) ] > 0. \begin{aligned}
2gP(1)P(\tau)-(1-\tau)(\alpha-\gamma\tau)(PQ'-QP') \\
&\ge g[2\alpha^2\tau+\alpha g(1-\tau)+\alpha\gamma(1-3\tau+6\tau^2)] \\
&>0.
\end{aligned} 2 g P ( 1 ) P ( τ ) − ( 1 − τ ) ( α − γ τ ) ( P Q ′ − Q P ′ ) ≥ g [ 2 α 2 τ + α g ( 1 − τ ) + α γ ( 1 − 3 τ + 6 τ 2 )] > 0. Division by P ( 1 ) P ( τ ) 2 P(1)P(\tau)^2 P ( 1 ) P ( τ ) 2 ( ω i − z i ) t i ≤ 2 C i / P (\omega_i-z_i)t_i\le2C_i/P ( ω i − z i ) t i ≤ 2 C i / P
For the second coordinate estimate, (113) shows that P R ′ − R P ′ PR'-RP' P R ′ − R P ′ τ = 1 \tau=1 τ = 1
α [ 4 g P ( 1 ) − ( P R ′ − R P ′ ) ( 1 ) ] = ( α − g − δ ) [ g ( β + 2 γ ) − ( α β − γ η ) ] + ( 2 α + η ) [ γ ( δ + g ) − ( α − g ) 2 ] + 2 g [ α g + γ ( δ + g ) ] + g ( δ + 2 g ) η + g ( 2 α + δ + g ) ( β − η ) > 0. (122) \begin{aligned}
\alpha[4gP(1)-(PR'-RP')(1)] \\
&=(\alpha-g-\delta)[g(\beta+2\gamma)-(\alpha\beta-\gamma\eta)] \\
&+(2\alpha+\eta)[\gamma(\delta+g)-(\alpha-g)^2] \\
&+2g[\alpha g+\gamma(\delta+g)]+g(\delta+2g)\eta\\
&+g(2\alpha+\delta+g)(\beta-\eta)>0. \tag*{(122)}
\end{aligned} α [ 4 g P ( 1 ) − ( P R ′ − R P ′ ) ( 1 )] = ( α − g − δ ) [ g ( β + 2 γ ) − ( α β − γ η )] + ( 2 α + η ) [ γ ( δ + g ) − ( α − g ) 2 ] + 2 g [ α g + γ ( δ + g )] + g ( δ + 2 g ) η + g ( 2 α + δ + g ) ( β − η ) > 0. (122) All terms are nonnegative by (119) and (120) , and 2 α g 2 > 0 2\alpha g^2>0 2 α g 2 > 0
ω i u i = P R ′ − R P ′ P ( τ ) P ( 1 ) ≤ 4 C i P ( τ ) . \omega_i u_i=\frac{PR'-RP'}{P(\tau)P(1)}\le\frac{4C_i}{P(\tau)}. ω i u i = P ( τ ) P ( 1 ) P R ′ − R P ′ ≤ P ( τ ) 4 C i . This proves both coordinate estimates and hence (115) when n = r + s n=r+s n = r + s s = 1 s=1 s = 1
Finally, suppose r + s > n r+s>n r + s > n r + s − n r+s-n r + s − n ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] z z z r + s r+s r + s ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 r = n r=n r = n P = σ n − 2 σ s − 1 > 0 P=\sigma_{n-2}\sigma_{s-1}>0 P = σ n − 2 σ s − 1 > 0 P ( z ( i ) ) P(z^{(i)}) P ( z ( i ) ) (115) in every stated dimension. □ \square □
Proof of the derivative estimate.
Proof of Lemma 6.2. Let λ = ( a , 1 , z ) \lambda=(a,1,z) λ = ( a , 1 , z ) a > 1 a>1 a > 1 r + s ≥ n r+s\ge n r + s ≥ n
σ s ∂ 1 σ r − σ r ∂ 1 σ s = P + Q + R , q 1 q 2 = 1 + t + u a 2 + a t + u . \sigma_s\partial_1\sigma_r-\sigma_r\partial_1\sigma_s=P+Q+R,\qquad\frac{q_1}{q_2}=\frac{1+t+u}{a^2+at+u}. σ s ∂ 1 σ r − σ r ∂ 1 σ s = P + Q + R , q 2 q 1 = a 2 + a t + u 1 + t + u . For brevity, set
U = a t + ( a + 1 ) u , S = 1 + t + u , T = a + 1 + t , C = a 2 + a t + u , H = a S ( 2 a + t ) + C ( t + 2 ) , K = a ( a + 1 ) t + a ( 3 a − 1 ) + ( a + 1 ) u . (123) \begin{aligned}
U&=at+(a+1)u,\qquad S=1+t+u,\qquad T=a+1+t,\qquad C=a^2+at+u,\\
H&=aS(2a+t)+C(t+2),\qquad K=a(a+1)t+a(3a-1)+(a+1)u. \tag*{(123)}
\end{aligned} U H = a t + ( a + 1 ) u , S = 1 + t + u , T = a + 1 + t , C = a 2 + a t + u , = a S ( 2 a + t ) + C ( t + 2 ) , K = a ( a + 1 ) t + a ( 3 a − 1 ) + ( a + 1 ) u . (123) Thus q 1 / q 2 = S / C q_1/q_2=S/C q 1 / q 2 = S / C D q [ v ] = 0 Dq[v]=0 D q [ v ] = 0
c i = − a q 2 ∂ i ( q 1 / q 2 ) . c_i=-aq_2\partial_i(q_1/q_2). c i = − a q 2 ∂ i ( q 1 / q 2 ) . Differentiating the quotient before setting ( λ 1 , λ 2 ) = ( a , 1 ) (\lambda_1,\lambda_2)=(a,1) ( λ 1 , λ 2 ) = ( a , 1 )
c 1 = a q 2 S ( 2 a + t ) C 2 , c 2 = − a q 2 ( t + 2 ) C , c i = − a q 2 ( a − 1 ) C 2 [ ( a − u ) t i + T u i ] , i ≥ 3. (124) \begin{aligned}
c_1&=\frac{aq_2S(2a+t)}{C^2},\qquad c_2=-\frac{aq_2(t+2)}{C},\\
c_i&=-\frac{aq_2(a-1)}{C^2}\left[(a-u)t_i+Tu_i\right],\qquad i\ge3. \tag*{(124)}
\end{aligned} c 1 c i = C 2 a q 2 S ( 2 a + t ) , c 2 = − C a q 2 ( t + 2 ) , = − C 2 a q 2 ( a − 1 ) [ ( a − u ) t i + T u i ] , i ≥ 3. (124) Consequently,
c 1 − c 2 a = q 2 H C 2 , − D 2 q [ v , v ] = 2 a 2 q 2 S T C 2 > 0 , 2 a − 1 ( q 1 − q 2 a 2 ) = 2 q 2 U a 2 C . (125) \begin{aligned}
c_1-\frac{c_2}{a}&=\frac{q_2H}{C^2},\\
-D^2q[v,v]&=\frac{2a^2q_2ST}{C^2}>0,\\
\frac{2}{a-1}\left(q_1-\frac{q_2}{a^2}\right)&=\frac{2q_2U}{a^2C}. \tag*{(125)}
\end{aligned} c 1 − a c 2 − D 2 q [ v , v ] a − 1 2 ( q 1 − a 2 q 2 ) = C 2 q 2 H , = C 2 2 a 2 q 2 S T > 0 , = a 2 C 2 q 2 U . (125) For i ≥ 3 i\ge3 i ≥ 3 σ s ∂ i σ r − σ r ∂ i σ s \sigma_s\partial_i\sigma_r-\sigma_r\partial_i\sigma_s σ s ∂ i σ r − σ r ∂ i σ s z i z_i z i z i z_i z i 1 1 1 t t t u u u
q i q 2 = P ( z ( i ) ) [ a 2 + a t ( z ( i ) ) + u ( z ( i ) ) ] P ( z ) C ≥ 1 ω i . \frac{q_i}{q_2}=\frac{P(z^{(i)})\left[a^2+at(z^{(i)})+u(z^{(i)})\right]}{P(z)C}\ge\frac{1}{\omega_i}. q 2 q i = P ( z ) C P ( z ( i ) ) [ a 2 + a t ( z ( i ) ) + u ( z ( i ) ) ] ≥ ω i 1 . It follows that
∑ i = 3 n a − z i 2 q i c i 2 ≤ a 3 q 2 ( a − 1 ) 2 2 C 4 ∑ i = 3 n ω i [ ( a − u ) t i + T u i ] 2 . (126) \sum_{i=3}^{n}\frac{a-z_i}{2q_i}c_i^2\le\frac{a^3q_2(a-1)^2}{2C^4}\sum_{i=3}^{n}\omega_i\left[(a-u)t_i+Tu_i\right]^2. \tag*{(126)} i = 3 ∑ n 2 q i a − z i c i 2 ≤ 2 C 4 a 3 q 2 ( a − 1 ) 2 i = 3 ∑ n ω i [ ( a − u ) t i + T u i ] 2 . (126) By Lemmas 6.3 and 6.4,
∑ i = 3 n ω i [ ( a − u ) t i + T u i ] 2 ≤ ( a − u ) 2 ( t + 2 ) 2 + 4 χ T 2 t + 8 ( a − u ) T , (127) \sum_{i=3}^{n}\omega_i\left[(a-u)t_i+Tu_i\right]^2\le(a-u)^2(t+2)^2+4\chi T^2t+8(a-u)T, \tag*{(127)} i = 3 ∑ n ω i [ ( a − u ) t i + T u i ] 2 ≤ ( a − u ) 2 ( t + 2 ) 2 + 4 χ T 2 t + 8 ( a − u ) T , (127) where χ = 1 \chi=1 χ = 1 r − s ≥ 2 r-s\ge2 r − s ≥ 2 χ = 2 \chi=2 χ = 2 r − s = 1 r-s=1 r − s = 1 t i 2 ≤ t i t_i^2\le t_i t i 2 ≤ t i u i 2 ≤ χ t u i u_i^2\le\chi tu_i u i 2 ≤ χ t u i t i u i ≤ u i t_iu_i\le u_i t i u i ≤ u i t < 2 t<2 t < 2 0 ≤ u ≤ 1 0\le u\le1 0 ≤ u ≤ 1
It remains to verify the scalar inequality
C ( H 2 − 4 U S T ) > a U ( a − 1 ) 2 ∑ i = 3 n ω i [ ( a − u ) t i + T u i ] 2 . (128) C(H^2-4UST)>aU(a-1)^2\sum_{i=3}^{n}\omega_i\left[(a-u)t_i+Tu_i\right]^2. \tag*{(128)} C ( H 2 − 4 U S T ) > a U ( a − 1 ) 2 i = 3 ∑ n ω i [ ( a − u ) t i + T u i ] 2 . (128) The key factorization is
H 2 − 4 U S T = 4 ( a − u ) S K T + ( a − 1 ) 2 ( a − u ) 2 ( t + 2 ) 2 . (129) H^2-4UST=4(a-u)SKT+(a-1)^2(a-u)^2(t+2)^2. \tag*{(129)} H 2 − 4 U S T = 4 ( a − u ) S K T + ( a − 1 ) 2 ( a − u ) 2 ( t + 2 ) 2 . (129) Moreover,
( t + 2 ) C − a U = a t 2 + 2 a t + ( 2 − u ) a 2 − u a + u ( t + 2 ) > 0. (t+2)C-aU=at^2+2at+(2-u)a^2-ua+u(t+2)>0. ( t + 2 ) C − a U = a t 2 + 2 a t + ( 2 − u ) a 2 − u a + u ( t + 2 ) > 0. since ( 2 − u ) a 2 − u a ≥ a 2 − a > 0 (2-u)a^2-ua\ge a^2-a>0 ( 2 − u ) a 2 − u a ≥ a 2 − a > 0 (129) , after multiplication by C C C (128) containing ( a − u ) 2 ( t + 2 ) (a-u)^2(t+2) ( a − u ) 2 ( t + 2 ) (127) , it is therefore enough to show
C ( a − u ) S K > a U ( a − 1 ) 2 [ χ T t + 2 ( a − u ) ] . (130) C(a-u)SK>aU(a-1)^2[\chi Tt+2(a-u)]. \tag*{(130)} C ( a − u ) S K > a U ( a − 1 ) 2 [ χ T t + 2 ( a − u )] . (130) If χ = 1 \chi=1 χ = 1
U ≤ a S , a − u ≥ a − 1 , C ≥ a ( a + t ) , K ≥ a [ ( a + 1 ) t + 3 a − 1 ] . U\le aS,\qquad a-u\ge a-1,\qquad C\ge a(a+t),\qquad K\ge a[(a+1)t+3a-1]. U ≤ a S , a − u ≥ a − 1 , C ≥ a ( a + t ) , K ≥ a [( a + 1 ) t + 3 a − 1 ] . Hence (6.32) follows from
( a + t ) [ ( a + 1 ) t + 3 a − 1 ] − ( a − 1 ) T t − 2 ( a − 1 ) 2 = 2 t 2 + 4 a t + a 2 + 3 a − 2 > 0. \begin{aligned}
(a+t)[(a+1)t+3a-1]-(a-1)Tt-2(a-1)^2 \\
&=2t^2+4at+a^2+3a-2>0.
\end{aligned} ( a + t ) [( a + 1 ) t + 3 a − 1 ] − ( a − 1 ) T t − 2 ( a − 1 ) 2 = 2 t 2 + 4 a t + a 2 + 3 a − 2 > 0. If χ = 2 \chi=2 χ = 2 0 ≤ t ≤ 2 0\le t\le2 0 ≤ t ≤ 2
U a S ≤ 1 − a − 1 a ( t + 2 ) , ( a − 1 ) 2 ( T t a − u + 1 ) ≤ a ( a + t ) ( t + 1 ) . \frac{U}{aS}\le1-\frac{a-1}{a(t+2)},\qquad(a-1)^2\left(\frac{Tt}{a-u}+1\right)\le a(a+t)(t+1). a S U ≤ 1 − a ( t + 2 ) a − 1 , ( a − 1 ) 2 ( a − u T t + 1 ) ≤ a ( a + t ) ( t + 1 ) . Furthermore,
( a + 1 ) t + 3 a − 1 − 2 ( 1 − a − 1 a ( t + 2 ) ) a ( t + 1 ) = ( a − 1 ) ( 4 + t − t 2 ) t + 2 > 0. (a+1)t+3a-1-2\left(1-\frac{a-1}{a(t+2)}\right)a(t+1)=\frac{(a-1)(4+t-t^2)}{t+2}>0. ( a + 1 ) t + 3 a − 1 − 2 ( 1 − a ( t + 2 ) a − 1 ) a ( t + 1 ) = t + 2 ( a − 1 ) ( 4 + t − t 2 ) > 0. Consequently,
C K a 2 ≥ ( a + t ) [ ( a + 1 ) t + 3 a − 1 ] > 2 U a S ( a − 1 ) 2 ( T t a − u + 1 ) , \frac{CK}{a^2}\ge(a+t)[(a+1)t+3a-1]>2\frac{U}{aS}(a-1)^2\left(\frac{Tt}{a-u}+1\right), a 2 C K ≥ ( a + t ) [( a + 1 ) t + 3 a − 1 ] > 2 a S U ( a − 1 ) 2 ( a − u T t + 1 ) , which proves (130) . Thus (128) holds, and Lemma 6.2 is proved. □
Acknowledgements
This work was supported by the National Key Research and Development Program of China (2021YFA1001800), National Natural Science Foundation of China (No. 12531002) and the Fundamental Research Funds for the Central Universities. The third author was supported by NSFC Grant No. 12601099. The last author was also supported by the China Postdoctoral Science Foundation (Grant No. 2025M783146).
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China
Email address: Luo_tianci@mail.ustc.edu.cn
Email address: yongwei@ustc.edu.cn
Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China
Email address: zhangrj76@mail.sysu.edu.cn
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China
Email address: zhourong@mail.ustc.edu.cn
