Introduction
A classical problem for contracting curvature flows is to determine whether closed convex hypersurfaces shrink to round points. Huisken [24 ] proved that every smooth closed strictly convex hypersurface in Euclidean space contracts to a round point under mean curvature flow. Alternative proofs were given by Hamilton [22 ] and Andrews [2 ] . Chow [15 ] established the corresponding result for the n n n [16 ] . Andrews [2 ] obtained convergence results for a large class of homogeneous degree-one curvature flows. In particular, his later pinching estimates [8 ] established round-point convergence for the speeds σ 1 / k \sigma^{1/k} σ 1/ k 1 ≤ k ≤ n 1 \leq k \leq n 1 ≤ k ≤ n
The Gauss curvature flow has been studied extensively, beginning with Firey’s model for the abrasion of convex bodies [17 ] . Tso [32 ] proved finite-time contraction to a point, and Andrews [6 ] established round-point convergence for convex surfaces in R 3 \mathbb{R}^3 R 3 [7 ] . In higher dimensions, Guan–Ni [21 ] proved smooth convergence of the volume-normalized Gauss curvature flow to a strictly convex self-similar solution. Andrews–Guan–Ni [10 ] obtained corresponding convergence results for K α K^\alpha K α α > 1 / ( n + 2 ) \alpha> 1/(n + 2) α > 1/ ( n + 2 ) [14 ] , these results give round-point convergence for α > 1 / ( n + 2 ) \alpha> 1/(n + 2) α > 1/ ( n + 2 ) α = 1 / ( n + 2 ) \alpha= 1/(n + 2) α = 1/ ( n + 2 ) [5 ] .
For other curvature speeds of homogeneity greater than one, convergence results have often required additional assumptions on the initial hypersurface. Schulze [31 ] proved round-point convergence for powers of the mean curvature under suitable initial curvature pinching assumptions. In the appendix, jointly with Schnürer, he also proved convergence without pinching for convex surfaces in R 3 \mathbb{R}^{3} R 3 H α H^\alpha H α 1 ≤ α ≤ 5 1 \le\alpha\le5 1 ≤ α ≤ 5 [11 ] obtained convergence results for more general curvature speeds under sufficiently strong initial pinching assumptions. For powers of the scalar curvature, Alessandroni–Sinestrari [1 ] proved round-point convergence for σ 2 α \sigma_{2}^{\alpha} σ 2 α α > 1 / 2 \alpha> 1/2 α > 1/2 [27 ] proved spherical convergence for the speeds σ k α \sigma_{k}^{\alpha} σ k α 1 / k ≤ α ≤ c ( n , k ) 1/k \le\alpha\le c(n,k) 1/ k ≤ α ≤ c ( n , k ) c ( n , k ) > 1 / k c(n,k) > 1/k c ( n , k ) > 1/ k σ k α \sigma_{k}^{\alpha} σ k α 2 ≤ k ≤ n − 1 2 \le k \le n - 1 2 ≤ k ≤ n − 1
Some related results are available without these assumptions. Guan–Huang–Liu [20 Theorem 1.1 and Remark 5.1(2)] proved finite-time contraction to a point for a class of curvature flows which includes σ k α \sigma_{k}^{\alpha} σ k α 1 ≤ k ≤ n 1 \le k \le n 1 ≤ k ≤ n α > 0 \alpha> 0 α > 0 [18 Theorem 1.4] proved that smooth closed strictly convex self-similar shrinkers for the σ k α \sigma_{k}^{\alpha} σ k α 1 ≤ k < n 1 \le k < n 1 ≤ k < n α ≥ 1 / k \alpha\ge1/k α ≥ 1/ k [19 ] . However, point extinction and the classification of self-similar shrinkers do not by themselves imply convergence of the rescaled flow. For this, one needs uniform estimates and a way to show that the normalized hypersurfaces approach a sphere.
The curvature pinching arguments used for degree-one flows do not extend directly to the speeds σ k α \sigma_{k}^{\alpha} σ k α 2 ≤ k ≤ n − 1 2 \le k \le n - 1 2 ≤ k ≤ n − 1 [1 ] , the higher homogeneity introduces gradient terms which are difficult to control without an initial pinching assumption. The entropy method for the Gauss curvature flow also uses features particular to Gauss curvature. In Gauss coordinates, K d μ = d σ K\,d\mu= d\sigma K d μ = d σ σ k d μ \sigma_{k}\,d\mu σ k d μ k < n k < n k < n
Related results have been obtained for constrained curvature flows. Andrews–Wei [13 ] proved smooth convergence to spheres for flows by positive powers of σ k \sigma_{k} σ k [12 ] and the monograph [9 ] .
In this paper, we prove round-point convergence for the σ k \sigma_{k} σ k 2 ≤ k ≤ n − 1 2 \le k \le n - 1 2 ≤ k ≤ n − 1
1.1. Main result. Let X 0 : M n → R n + 1 X_{0}: M^{n} \to\mathbb{R}^{n+1} X 0 : M n → R n + 1 M 0 M_{0} M 0 κ = ( κ 1 , … , κ n ) \kappa= (\kappa_{1}, \ldots, \kappa_{n}) κ = ( κ 1 , … , κ n )
σ k ( κ ) = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n κ i 1 ⋯ κ i k , E k ( κ ) = ( n k ) − 1 σ k ( κ ) . (1) \sigma_k(\kappa)=\sum_{1\le i_1<\cdots<i_k\le n}\kappa_{i_1}\cdots\kappa_{i_k},\qquad E_k(\kappa)=\binom{n}{k}^{-1}\sigma_k(\kappa). \tag*{(1)} σ k ( κ ) = 1 ≤ i 1 < ⋯ < i k ≤ n ∑ κ i 1 ⋯ κ i k , E k ( κ ) = ( k n ) − 1 σ k ( κ ) . (1) In particular, σ 1 = H \sigma_1=H σ 1 = H σ n = K \sigma_n=K σ n = K H H H K K K E 1 = H / n E_1=H/n E 1 = H / n E n = K E_n=K E n = K
We consider the contracting σ k \sigma_k σ k M t = X ( M , t ) M_t=X(M,t) M t = X ( M , t ) E k E_k E k
{ ∂ t X = − E k ( W ) ν , X ( ⋅ , 0 ) = X 0 , (2) \begin{cases}
\partial_tX=-E_k(W)\nu,\\
X(\cdot,0)=X_0, \tag*{(2)}
\end{cases} { ∂ t X = − E k ( W ) ν , X ( ⋅ , 0 ) = X 0 , (2) where X ( ⋅ , t ) X(\cdot,t) X ( ⋅ , t ) ν \nu ν W W W M t M_t M t E k = ( n k ) − 1 σ k E_k=\binom{n}{k}^{-1}\sigma_k E k = ( k n ) − 1 σ k Ω t \Omega_t Ω t M t M_t M t h ( ⋅ , t ) h(\cdot,t) h ( ⋅ , t )
Theorem 1.1. Let n ≥ 3 n\ge3 n ≥ 3 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 M 0 ⊂ R n + 1 M_0\subset\mathbb{R}^{n+1} M 0 ⊂ R n + 1 (2) has a unique maximal smooth strictly convex solution on [ 0 , T ) [0,T) [ 0 , T ) T < ∞ T<\infty T < ∞ M t M_t M t x ∗ ∈ R n + 1 x_*\in\mathbb{R}^{n+1} x ∗ ∈ R n + 1
Let b ( t ) b(t) b ( t ) Ω t \Omega_t Ω t
b ( t ) = 1 ∣ S n ∣ ∫ S n h ( z , t ) d σ , τ ( t ) = ∫ 0 t b ( s ) − ( k + 1 ) d s , b(t)=\frac{1}{\lvert\mathbb{S}^n\rvert}\int_{\mathbb{S}^n}h(z,t)\,d\sigma,\qquad\tau(t)=\int_0^t b(s)^{-(k+1)}\,ds, b ( t ) = ∣ S n ∣ 1 ∫ S n h ( z , t ) d σ , τ ( t ) = ∫ 0 t b ( s ) − ( k + 1 ) d s , Y ( ⋅ , τ ( t ) ) = X ( ⋅ , t ) − x ∗ b ( t ) . (3) Y(\cdot,\tau(t))=\frac{X(\cdot,t)-x_*}{b(t)}. \tag*{(3)} Y ( ⋅ , τ ( t )) = b ( t ) X ( ⋅ , t ) − x ∗ . (3) Then τ ( t ) → ∞ \tau(t)\to\infty τ ( t ) → ∞ t → T t\to T t → T u u u Y Y Y
u ( ⋅ , τ ) ⟶ 1 in C j ( S n ) for every integer j ≥ 0. (4) u(\cdot,\tau)\longrightarrow1\quad\text{in }C^j(\mathbb{S}^n)\quad\text{for every integer }j\ge0. \tag*{(4)} u ( ⋅ , τ ) ⟶ 1 in C j ( S n ) for every integer j ≥ 0. (4) The convergence is exponential in the normalized time.
There are c , C > 0 c,C>0 c , C > 0 M 0 , n , k M_0,n,k M 0 , n , k
c ( T − t ) − 1 / ( k + 1 ) ≤ κ i ( M t ) ≤ C ( T − t ) − 1 / ( k + 1 ) , 1 ≤ i ≤ n , (5) c(T-t)^{-1/(k+1)}\le\kappa_i(M_t)\le C(T-t)^{-1/(k+1)},\qquad1\le i\le n, \tag*{(5)} c ( T − t ) − 1/ ( k + 1 ) ≤ κ i ( M t ) ≤ C ( T − t ) − 1/ ( k + 1 ) , 1 ≤ i ≤ n , (5) for all t t t T T T H H H E k E_k E k
The existence and point-extinction assertions follow from Proposition 2.2 . The normalization fixes the mean width at two. The exponential convergence rate is given in Corollary 5.3.
Theorem 1.1 establishes round-point convergence for the flow (2) in the intermediate range 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 Corollary 4.2 ) and yields a quantitative stability estimate (Proposition 4.6 ).
Discussion of the proof
The main difficulty for the intermediate flows 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1
A strengthened Codazzi inequality. Let σ \sigma σ ∇ ‾ \overline{\nabla} ∇ S n \mathbb{S}^n S n B = ∇ ‾ 2 h + h σ B=\overline{\nabla}^2h+h\sigma B = ∇ 2 h + hσ B ~ = ∇ ‾ 2 u + u σ \widetilde{B}=\overline{\nabla}^{2}u+u\sigma B = ∇ 2 u + u σ G ( B ) = − E k ( B − 1 ) = − F G(B)=-E_k(B^{-1})=-F G ( B ) = − E k ( B − 1 ) = − F ∇ ‾ B \overline{\nabla}B ∇ B 1 ≤ k < n 1\le k<n 1 ≤ k < n
− G ¨ i j , r s T m i j T m r s ≥ 2 n E 1 ( B ) G ¨ i j T i p p T j q q + 1 F ∑ m ( G ¨ i j T m i j ) 2 -\ddot{G}^{ij,rs}T_{mij}T_{mrs}\ge\frac{2}{nE_1(B)}\ddot{G}^{ij}T_{ipp}T_{jqq}+\frac{1}{F}\sum_m\left(\ddot{G}^{ij}T_{mij}\right)^2 − G ¨ ij , r s T mij T m r s ≥ n E 1 ( B ) 2 G ¨ ij T i pp T j q q + F 1 m ∑ ( G ¨ ij T mij ) 2 for every fully symmetric three-tensor T T T T = 0 T=0 T = 0
Taking T = ∇ ‾ B T=\overline{\nabla}B T = ∇ B C G C_G C G (26) satisfies
C G ≥ ∣ ∇ ‾ F ∣ 2 F , C G = 0 ⟺ ∇ ‾ B = 0. C_G\ge\frac{\lvert\overline{\nabla}F\rvert^2}{F},\qquad C_G=0\quad\Longleftrightarrow\quad\overline{\nabla}B=0. C G ≥ F ∣ ∇ F ∣ 2 , C G = 0 ⟺ ∇ B = 0. Thus the inequality controls the gradient of the speed and characterizes the equality case.
A monotone entropy. For 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1
p = n − 1 k − 1 , q = p − 1 = n − k k − 1 , (6) p=\frac{n-1}{k-1},\qquad q=p-1=\frac{n-k}{k-1}, \tag*{(6)} p = k − 1 n − 1 , q = p − 1 = k − 1 n − k , (6) and define
E ( M ) = 1 ∣ S n ∣ ∫ M E k ( W ) p E 1 ( B ) q d μ . (7) \mathcal{E}(M)=\frac{1}{\lvert S^n\rvert}\int_M E_k(W)^p E_1(B)^q\,d\mu. \tag*{(7)} E ( M ) = ∣ S n ∣ 1 ∫ M E k ( W ) p E 1 ( B ) q d μ . (7) This entropy is invariant under translations and dilations. It has a geometric interpretation in terms of orthogonal projections. Indeed, writing j = n − k j=n-k j = n − k
E ( ∂ Ω ) = 1 ∣ S j ∣ ∫ Gr ( n + 1 , j + 1 ) ∫ ∂ ( Ω ∣ Π ) ( F E 1 ( B ) ) q ( ν Π ) d μ Π d Π . (8) \mathcal{E}(\partial\Omega)=\frac{1}{\lvert S^j\rvert}\int_{\operatorname{Gr}(n+1,j+1)}\int_{\partial(\Omega\mid\Pi)}\left(FE_1(B)\right)^q(\nu_\Pi)\,d\mu_\Pi\,d\Pi. \tag*{(8)} E ( ∂ Ω ) = ∣ S j ∣ 1 ∫ Gr ( n + 1 , j + 1 ) ∫ ∂ ( Ω ∣ Π ) ( F E 1 ( B ) ) q ( ν Π ) d μ Π d Π. (8) Here Ω ∣ Π \Omega\mid\Pi Ω ∣ Π Ω \Omega Ω Π \Pi Π d Π d\Pi d Π Gr ( n + 1 , j + 1 ) \operatorname{Gr}(n+1,j+1) Gr ( n + 1 , j + 1 ) ν Π \nu_\Pi ν Π d μ Π d\mu_\Pi d μ Π ν Π \nu_\Pi ν Π j j j F E 1 ( B ) FE_1(B) F E 1 ( B ) 1 − k 1-k 1 − k q = j / ( k − 1 ) q=j/(k-1) q = j / ( k − 1 ) p = q + 1 p=q+1 p = q + 1
The Newton–Maclaurin inequalities and the strengthened Codazzi inequality give
E ( M ) ≥ 1 , d d t E ( M t ) ≤ 0 , \mathcal{E}(M)\ge1,\qquad\frac{d}{dt}\mathcal{E}(M_t)\le0, E ( M ) ≥ 1 , d t d E ( M t ) ≤ 0 , with equality in either statement only for round spheres. The entropy also controls the shape of a convex body. If b b b Ω \Omega Ω
r − ( Ω ) ≥ c ( n , k ) b E ( ∂ Ω ) − ( k − 1 ) , diam Ω ≤ C ( n ) b . r_-(\Omega)\ge c(n,k)b\mathcal{E}(\partial\Omega)^{-(k-1)},\qquad\operatorname{diam}\Omega\le C(n)b. r − ( Ω ) ≥ c ( n , k ) b E ( ∂ Ω ) − ( k − 1 ) , diam Ω ≤ C ( n ) b . Thus an upper bound for the entropy prevents collapse after fixing the mean width. The inradius estimate follows from Cauchy’s projection formula and Hölder’s inequality. For earlier entropy estimates for curvature flows, see Andrews [3 , 7 ] and Hamilton [23 ] .
We also obtain a quantitative stability estimate. If c c c Ω \Omega Ω
d H ( Ω , B b ( c ) ‾ ) b ≤ C ( n , k ) ( E ( ∂ Ω ) − 1 ) 2 / ( n + 2 ) . \frac{d_H(\Omega,\overline{B_b(c)})}{b}\le C(n,k)\left(\mathcal{E}(\partial\Omega)-1\right)^{2/(n+2)}. b d H ( Ω , B b ( c ) ) ≤ C ( n , k ) ( E ( ∂ Ω ) − 1 ) 2/ ( n + 2 ) . This is an independent geometric consequence of the entropy and is not needed for the convergence argument.
Convergence of the normalized flow. We normalize the mean support function to one as in (3) . The entropy bound gives uniform control of the inradius and diameter. Together with the Harnack estimate and a maximum principle, this yields uniform upper bounds for the normalized speed and its product with the mean principal radius. Write F = E k ( B ~ − 1 ) F = E_k(\widetilde{B}^{-1}) F = E k ( B − 1 ) F ‾ = ∫ S n F d σ \overline{F} = \int_{\mathbb{S}^n} F\,d\sigma F = ∫ S n F d σ D D D (64) , the entropy formula and the strengthened Codazzi inequality give
d d τ E ( Y ( τ ) ) ≤ − D ( τ ) , D ( τ ) ≥ c ∫ S n ∣ ∇ ‾ F ∣ 2 d σ ≥ c ∫ S n ∣ F − F ‾ ∣ 2 d σ . \frac{d}{d\tau}\mathcal{E}(Y(\tau)) \leq-D(\tau), \qquad D(\tau) \geq c\int_{\mathbb{S}^n} \lvert\overline{\nabla}F\rvert^2\,d\sigma\geq c\int_{\mathbb{S}^n} \lvert F-\overline{F}\rvert^2\,d\sigma. d τ d E ( Y ( τ )) ≤ − D ( τ ) , D ( τ ) ≥ c ∫ S n ∣ ∇ F ∣ 2 d σ ≥ c ∫ S n ∣ F − F ∣ 2 d σ . On the other hand, the normalized support function satisfies
∂ τ ( u − 1 ) = F ‾ ( u − 1 ) − ( F − F ‾ ) , F ‾ ≥ 1. \partial_\tau(u-1)=\overline{F}(u-1)-(F-\overline{F}), \qquad\overline{F}\geq1. ∂ τ ( u − 1 ) = F ( u − 1 ) − ( F − F ) , F ≥ 1. Integrating this equation towards future times gives
∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s ⟶ 0 , \int_{\mathbb{S}^n}\lvert u(z,\tau)-1\rvert^2\,d\sigma\leq C\int_\tau^\infty e^{-(s-\tau)}D(s)\,ds\longrightarrow0, ∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s ⟶ 0 , since
∫ 0 ∞ D ( s ) d s ≤ E ( M 0 ) − 1. \int_0^\infty D(s)\,ds\leq\mathcal{E}(M_0)-1. ∫ 0 ∞ D ( s ) d s ≤ E ( M 0 ) − 1. Convexity and the diameter bound give ∣ ∇ ‾ u ∣ ≤ C \lvert\overline{\nabla}u\rvert\leq C ∣ ∇ u ∣ ≤ C u u u u ≥ 1 / 2 u\geq1/2 u ≥ 1/2 log ( F u k + 1 ) \log(Fu^{k+1}) log ( F u k + 1 ) F F F
c σ ≤ B ~ ≤ C σ . c\sigma\leq\widetilde{B}\leq C\sigma. c σ ≤ B ≤ C σ . Parabolic regularity now gives smooth convergence to the unit sphere. Linearization at the sphere, together with control of the translation modes, gives exponential convergence as stated in Corollary 5.3. Rescaling yields the curvature estimates in Theorem 1.1 .
The paper is organized as follows. Section 2 collects the preliminary results. Section 3 proves the Codazzi inequality. Section 4 establishes the entropy and its geometric estimates. Section 5 completes the convergence proof.
1.3. AI Declaration. During the preparation of this paper, the authors used OpenAI’s ChatGPT (6 Astra) to assist with exploring proof strategies, searching the literature, developing and checking mathematical arguments and calculations, and improving the exposition. The authors critically evaluated the suggestions, checked the mathematical arguments and cited sources, and revised the manuscript accordingly. The authors take full responsibility for all mathematical statements, proofs, references, and conclusions in the final manuscript.
Preliminaries
Throughout this section, n ≥ 2 n\geq2 n ≥ 2 1 ≤ k ≤ n 1\leq k\leq n 1 ≤ k ≤ n n n n
Elementary symmetric functions
For λ ∈ R n \lambda\in\mathbb{R}^n λ ∈ R n 0 ≤ j ≤ n 0\leq j\leq n 0 ≤ j ≤ n
σ j ( λ ) = ∑ 1 ≤ i 1 < ⋯ < i j ≤ n λ i 1 ⋯ λ i j , E j = σ j ( n j ) . \sigma_j(\lambda)=\sum_{1\leq i_1<\cdots<i_j\leq n}\lambda_{i_1}\cdots\lambda_{i_j},\qquad E_j=\frac{\sigma_j}{\binom{n}{j}}. σ j ( λ ) = 1 ≤ i 1 < ⋯ < i j ≤ n ∑ λ i 1 ⋯ λ i j , E j = ( j n ) σ j . We set σ 0 = E 0 = 1 \sigma_0=E_0=1 σ 0 = E 0 = 1 σ j = E j = 0 \sigma_j=E_j=0 σ j = E j = 0 j < 0 j<0 j < 0 j > n j>n j > n E j E_j E j
Γ + = { λ ∈ R n : λ i > 0 for all i } . \Gamma_+=\{\lambda\in\mathbb{R}^n:\lambda_i>0\text{ for all }i\}. Γ + = { λ ∈ R n : λ i > 0 for all i } . Writing λ ∣ i \lambda|i λ ∣ i λ ∣ i , l \lambda|i,l λ ∣ i , l
∂ σ j ∂ λ i = σ j − 1 ( λ ∣ i ) , ∂ 2 σ j ∂ λ i ∂ λ l = { σ j − 2 ( λ ∣ i , l ) , i ≠ l , 0 , i = l . (9) \frac{\partial\sigma_j}{\partial\lambda_i}=\sigma_{j-1}(\lambda|i),\qquad
\frac{\partial^2\sigma_j}{\partial\lambda_i\partial\lambda_l}=
\begin{cases}
\sigma_{j-2}(\lambda|i,l),&i\ne l,\\
0,&i=l.
\end{cases} \tag*{(9)} ∂ λ i ∂ σ j = σ j − 1 ( λ ∣ i ) , ∂ λ i ∂ λ l ∂ 2 σ j = { σ j − 2 ( λ ∣ i , l ) , 0 , i = l , i = l . (9) In particular,
∑ i λ i ∂ σ j ∂ λ i = j σ j , ∑ i λ i 2 ∂ σ j ∂ λ i = σ 1 σ j − ( j + 1 ) σ j + 1 . (10) \sum_i\lambda_i\frac{\partial\sigma_j}{\partial\lambda_i}=j\sigma_j,\qquad
\sum_i\lambda_i^2\frac{\partial\sigma_j}{\partial\lambda_i}
=\sigma_1\sigma_j-(j+1)\sigma_{j+1}. \tag*{(10)} i ∑ λ i ∂ λ i ∂ σ j = j σ j , i ∑ λ i 2 ∂ λ i ∂ σ j = σ 1 σ j − ( j + 1 ) σ j + 1 . (10) For a positive definite symmetric matrix B B B
E j ( B − 1 ) = E n − j ( B ) det B , 0 ≤ j ≤ n . (11) E_j(B^{-1})=\frac{E_{n-j}(B)}{\det B},\qquad0\le j\le n. \tag*{(11)} E j ( B − 1 ) = det B E n − j ( B ) , 0 ≤ j ≤ n . (11) A smooth symmetric function Φ \Phi Φ Γ + \Gamma_+ Γ +
d d s Φ ( B + s V ) ∣ s = 0 = Φ ˙ i j V i j , d 2 d s 2 Φ ( B + s V ) ∣ s = 0 = Φ ¨ i j , p q V i j V p q . \left.\frac{d}{ds}\Phi(B+sV)\right|_{s=0}=\dot{\Phi}^{ij}V_{ij},\qquad
\left.\frac{d^2}{ds^2}\Phi(B+sV)\right|_{s=0}
=\ddot{\Phi}^{ij,pq}V_{ij}V_{pq}. d s d Φ ( B + s V ) s = 0 = Φ ˙ ij V ij , d s 2 d 2 Φ ( B + s V ) s = 0 = Φ ¨ ij , pq V ij V pq . Write Φ i \Phi_i Φ i Φ i j \Phi_{ij} Φ ij B = diag ( r 1 , … , r n ) B=\operatorname{diag}(r_1,\ldots,r_n) B = diag ( r 1 , … , r n ) [8 Theorem 5.1] reads
Φ ˙ i j = Φ i δ i j , Φ ¨ i j , p q V i j V p q = ∑ i , j Φ i j V i i V j j + 2 ∑ i < j Φ i − Φ j r i − r j V i j 2 . (12) \dot{\Phi}^{ij}=\Phi_i\delta_{ij},\qquad
\ddot{\Phi}^{ij,pq}V_{ij}V_{pq}
=\sum_{i,j}\Phi_{ij}V_{ii}V_{jj}
+2\sum_{i<j}\frac{\Phi_i-\Phi_j}{r_i-r_j}V_{ij}^2. \tag*{(12)} Φ ˙ ij = Φ i δ ij , Φ ¨ ij , pq V ij V pq = i , j ∑ Φ ij V ii V j j + 2 i < j ∑ r i − r j Φ i − Φ j V ij 2 . (12) The divided differences extend continuously to repeated eigenvalues, with value Φ i i − Φ i j \Phi_{ii}-\Phi_{ij} Φ ii − Φ ij r i = r j r_i=r_j r i = r j
Convex hypersurfaces and evolution equations
Let M = ∂ Ω ⊂ R n + 1 M=\partial\Omega\subset\mathbb{R}^{n+1} M = ∂ Ω ⊂ R n + 1 ν \nu ν g g g ∇ \nabla ∇ h i j h_{ij} h ij d μ d\mu d μ D D D
h i j = ⟨ D e i ν , e j ⟩ , W = ( h i j ) , h i j = g i l h l j , H = tr W . h_{ij}=\langle D_{e_i}\nu,e_j\rangle,\qquad
W=(h^i{}_j),\qquad
h^i{}_j=g^{il}h_{lj},\qquad
H=\operatorname{tr}W. h ij = ⟨ D e i ν , e j ⟩ , W = ( h i j ) , h i j = g i l h l j , H = tr W . The eigenvalues κ i \kappa_i κ i W W W
F = E k ( W ) = E k ( κ ) . (13) F=E_k(W)=E_k(\kappa). \tag*{(13)} F = E k ( W ) = E k ( κ ) . (13) Let σ \sigma σ ∇ ‾ \overline{\nabla} ∇ d σ d\sigma d σ S n \mathbb{S}^n S n
∫ S n v d σ = 1 ∣ S n ∣ ∫ S n v d σ . \int_{\mathbb{S}^n}v\,d\sigma=\frac{1}{|\mathbb{S}^n|}\int_{\mathbb{S}^n}v\,d\sigma. ∫ S n v d σ = ∣ S n ∣ 1 ∫ S n v d σ . The support function and inverse Gauss map are
h ( z ) = sup x ∈ Ω ⟨ x , z ⟩ , X ( z ) = ∇ ‾ h ( z ) + h ( z ) z . (14) h(z)=\sup_{x\in\Omega}\langle x,z\rangle,\qquad
X(z)=\overline{\nabla}h(z)+h(z)z. \tag*{(14)} h ( z ) = x ∈ Ω sup ⟨ x , z ⟩ , X ( z ) = ∇ h ( z ) + h ( z ) z . (14) Identifying both tangent spaces with z ⊥ z^\perp z ⊥
B i j = ∇ ‾ i ∇ ‾ j h + h σ i j , W = B − 1 , E 1 ( B ) = 1 n tr σ B . (15) B_{ij}=\overline{\nabla}_i\overline{\nabla}_j h+h\sigma_{ij},\qquad
W=B^{-1},\qquad
E_1(B)=\frac{1}{n}\operatorname{tr}_{\sigma}B. \tag*{(15)} B ij = ∇ i ∇ j h + h σ ij , W = B − 1 , E 1 ( B ) = n 1 tr σ B . (15) Its eigenvalues r i = κ i − 1 r_i=\kappa_i^{-1} r i = κ i − 1 B B B σ \sigma σ
d μ = det B d σ , E 1 ( B ) = E n − 1 ( W ) E n ( W ) , F = E n − k ( B ) det B . (16) d\mu=\det B\,d\sigma,\qquad
E_1(B)=\frac{E_{n-1}(W)}{E_n(W)},\qquad
F=\frac{E_{n-k}(B)}{\det B}. \tag*{(16)} d μ = det B d σ , E 1 ( B ) = E n ( W ) E n − 1 ( W ) , F = det B E n − k ( B ) . (16) The Codazzi equation
∇ ‾ i B j l = ∇ ‾ j B i l (17) \overline{\nabla}_iB_{jl}=\overline{\nabla}_jB_{il} \tag*{(17)} ∇ i B j l = ∇ j B i l (17) makes ∇ ˉ B \bar{\nabla}B ∇ ˉ B
∇ ˉ i ( ∂ E j ( B ) ∂ B i l ) = 0 ( 1 ≤ j ≤ n ) . (18) \bar{\nabla}_{i}\left(\frac{\partial E_j(B)}{\partial B_{il}}\right)=0 \qquad(1\le j\le n). \tag*{(18)} ∇ ˉ i ( ∂ B i l ∂ E j ( B ) ) = 0 ( 1 ≤ j ≤ n ) . (18) This includes the cofactor tensor ( det B ) B − 1 (\det B)B^{-1} ( det B ) B − 1
In the normal parametrization ∂ t X = − F ν \partial_tX=-F\nu ∂ t X = − F ν
∂ t g i j = − 2 F h i j . (19) \partial_tg_{ij}=-2Fh_{ij}. \tag*{(19)} ∂ t g ij = − 2 F h ij . (19) See [24 Section 3] and [12 Lemma 9] . In Gauss coordinates, the support equation is
∂ t h = G ( B ) , G ( B ) = − E k ( B − 1 ) = − F . (20) \partial_th=G(B),\qquad G(B)=-E_k(B^{-1})=-F. \tag*{(20)} ∂ t h = G ( B ) , G ( B ) = − E k ( B − 1 ) = − F . (20) We write
G ˙ i j = ∂ G ∂ B i j , G ¨ i j , p q = ∂ 2 G ∂ B i j ∂ B p q . (21) \dot{G}^{ij}=\frac{\partial G}{\partial B_{ij}},\qquad\ddot{G}^{ij,pq}=\frac{\partial^2G}{\partial B_{ij}\partial B_{pq}}. \tag*{(21)} G ˙ ij = ∂ B ij ∂ G , G ¨ ij , pq = ∂ B ij ∂ B pq ∂ 2 G . (21) These derivatives are evaluated at the radius tensor under consideration, with σ \sigma σ E k 1 / k E_k^{1/k} E k 1/ k [12 Section 2] ), the function E k ( B − 1 ) − 1 / k E_k(B^{-1})^{-1/k} E k ( B − 1 ) − 1/ k s ↦ − s − k s\mapsto-s^{-k} s ↦ − s − k s > 0 s>0 s > 0 G ( B ) = − E k ( B − 1 ) G(B)=-E_k(B^{-1}) G ( B ) = − E k ( B − 1 ) ( G ˙ i j ) (\dot{G}^{ij}) ( G ˙ ij ) G ˙ i j B i j = k F \dot{G}^{ij}B_{ij}=kF G ˙ ij B ij = k F
∂ t X = − F z − ∇ ˉ F . (22) \partial_tX=-Fz-\bar{\nabla}F. \tag*{(22)} ∂ t X = − F z − ∇ ˉ F . (22) Since d X = B dX=B d X = B − ∇ ˉ F = − B ∇ F -\bar{\nabla}F=-B\nabla F − ∇ ˉ F = − B ∇ F
Lemma 2.1. Along the original flow (2) , with time derivatives taken at fixed Gauss normal,
( ∂ t − G ˙ i j ∇ ˉ i ∇ ˉ j ) h = − ( k + 1 ) F + ( G ˙ i j σ i j ) h , (23) (\partial_t-\dot{G}^{ij}\bar{\nabla}_i\bar{\nabla}_j)h=-(k+1)F+(\dot{G}^{ij}\sigma_{ij})h, \tag*{(23)} ( ∂ t − G ˙ ij ∇ ˉ i ∇ ˉ j ) h = − ( k + 1 ) F + ( G ˙ ij σ ij ) h , (23) ( ∂ t − G ˙ i j ∇ ˉ i ∇ ˉ j ) F = ( G ˙ i j σ i j ) F , (24) (\partial_t-\dot{G}^{ij}\bar{\nabla}_i\bar{\nabla}_j)F=(\dot{G}^{ij}\sigma_{ij})F, \tag*{(24)} ( ∂ t − G ˙ ij ∇ ˉ i ∇ ˉ j ) F = ( G ˙ ij σ ij ) F , (24) ( ∂ t − G ˙ p q ∇ ˉ p ∇ ˉ q ) B i j = G ¨ p q , r s ∇ ˉ i B p q ∇ ˉ j B r s + ( k − 1 ) F σ i j − ( G ˙ p q σ p q ) B i j . (25) (\partial_t-\dot{G}^{pq}\bar{\nabla}_p\bar{\nabla}_q)B_{ij}=\ddot{G}^{pq,rs}\bar{\nabla}_iB_{pq}\bar{\nabla}_jB_{rs}+(k-1)F\sigma_{ij}-(\dot{G}^{pq}\sigma_{pq})B_{ij}. \tag*{(25)} ( ∂ t − G ˙ pq ∇ ˉ p ∇ ˉ q ) B ij = G ¨ pq , r s ∇ ˉ i B pq ∇ ˉ j B r s + ( k − 1 ) F σ ij − ( G ˙ pq σ pq ) B ij . (25) Define
C G = − G ¨ i j , p q ∇ ˉ m B i j ∇ ˉ m B p q − 2 n E 1 ( B ) G ˙ i j ∇ ˉ i E 1 ( B ) ∇ ˉ j E 1 ( B ) . (26) C_G=-\ddot{G}^{ij,pq}\bar{\nabla}_mB_{ij}\bar{\nabla}_mB_{pq}-\frac{2n}{E_1(B)}\dot{G}^{ij}\bar{\nabla}_iE_1(B)\bar{\nabla}_jE_1(B). \tag*{(26)} C G = − G ¨ ij , pq ∇ ˉ m B ij ∇ ˉ m B pq − E 1 ( B ) 2 n G ˙ ij ∇ ˉ i E 1 ( B ) ∇ ˉ j E 1 ( B ) . (26) Then
( ∂ t − G ˙ i j ∇ ˉ i ∇ ˉ j ) E 1 ( B ) = − 2 E 1 ( B ) G ˙ i j ∇ ˉ i E 1 ( B ) ∇ ˉ j E 1 ( B ) + ( k − 1 ) F − ( G ˙ i j σ i j ) E 1 ( B ) − 1 n C G . (27) (\partial_t-\dot{G}^{ij}\bar{\nabla}_i\bar{\nabla}_j)E_1(B)
=-\frac{2}{E_1(B)}\dot{G}^{ij}\bar{\nabla}_iE_1(B)\bar{\nabla}_jE_1(B)
+(k-1)F-(\dot{G}^{ij}\sigma_{ij})E_1(B)-\frac{1}{n}C_G. \tag*{(27)} ( ∂ t − G ˙ ij ∇ ˉ i ∇ ˉ j ) E 1 ( B ) = − E 1 ( B ) 2 G ˙ ij ∇ ˉ i E 1 ( B ) ∇ ˉ j E 1 ( B ) + ( k − 1 ) F − ( G ˙ ij σ ij ) E 1 ( B ) − n 1 C G . (27) and
( ∂ t − G ˙ i j ∇ ˉ i ∇ ˉ j ) ( F E 1 ( B ) ) = − 2 G ˙ i j ∇ ˉ i log E 1 ( B ) ∇ ˉ j ( F E 1 ( B ) ) + ( k − 1 ) F 2 − F n C G . (28) (\partial_t-\dot{G}^{ij}\bar{\nabla}_i\bar{\nabla}_j)(FE_1(B))
=-2\dot{G}^{ij}\bar{\nabla}_i\log E_1(B)\bar{\nabla}_j(FE_1(B))+(k-1)F^2-\frac{F}{n}C_G. \tag*{(28)} ( ∂ t − G ˙ ij ∇ ˉ i ∇ ˉ j ) ( F E 1 ( B )) = − 2 G ˙ ij ∇ ˉ i log E 1 ( B ) ∇ ˉ j ( F E 1 ( B )) + ( k − 1 ) F 2 − n F C G . (28) Proof. The support equation and homogeneity give (23) and (24) . Differentiating (15) gives ∂ t B i j = − ∇ ˉ i ∇ ˉ j F − F σ i j \partial_tB_{ij}=-\bar{\nabla}_i\bar{\nabla}_jF-F\sigma_{ij} ∂ t B ij = − ∇ ˉ i ∇ ˉ j F − F σ ij S n S^n S n (17) , we obtain
G ˙ p q ∇ ˉ i ∇ ˉ j B p q = G ˙ p q ∇ ˉ p ∇ ˉ q B i j + k F σ i j − ( G ˙ p q σ p q ) B i j . \dot{G}^{pq}\bar{\nabla}_i\bar{\nabla}_jB_{pq}
=\dot{G}^{pq}\bar{\nabla}_p\bar{\nabla}_qB_{ij}+kF\sigma_{ij}-(\dot{G}^{pq}\sigma_{pq})B_{ij}. G ˙ pq ∇ ˉ i ∇ ˉ j B pq = G ˙ pq ∇ ˉ p ∇ ˉ q B ij + k F σ ij − ( G ˙ pq σ pq ) B ij . Together with
∇ ˉ i ∇ ˉ j F = − G ˙ p q ∇ ˉ i ∇ ˉ j B p q − G ¨ p q , r s ∇ ˉ i B p q ∇ ˉ j B r s , \bar{\nabla}_i\bar{\nabla}_jF=-\dot{G}^{pq}\bar{\nabla}_i\bar{\nabla}_jB_{pq}-\ddot{G}^{pq,rs}\bar{\nabla}_iB_{pq}\bar{\nabla}_jB_{rs}, ∇ ˉ i ∇ ˉ j F = − G ˙ pq ∇ ˉ i ∇ ˉ j B pq − G ¨ pq , r s ∇ ˉ i B pq ∇ ˉ j B r s , this proves (25) . Compare [12 Lemma 10] . Taking the trace gives (27) , and the product rule gives (28) . □
Point extinction and the Harnack estimate
We recall the point-extinction theorem of Guan–Huang–Liu [20 Theorem 1.1 and Remark 5.1(2)] .
Proposition 2.2. Let M 0 ⊂ R n + 1 M_0 \subset\mathbb{R}^{n+1} M 0 ⊂ R n + 1 (2) has a unique maximal smooth strictly convex solution on [ 0 , T ) [0,T) [ 0 , T ) T < ∞ T<\infty T < ∞ M t M_t M t x ∗ ∈ R n + 1 x_*\in\mathbb{R}^{n+1} x ∗ ∈ R n + 1
r − ( Ω 0 ) k + 1 k + 1 ≤ T ≤ r + ( Ω 0 ) k + 1 k + 1 , (29) \frac{r_-(\Omega_0)^{k+1}}{k+1}\leq T\leq\frac{r_+(\Omega_0)^{k+1}}{k+1}, \tag*{(29)} k + 1 r − ( Ω 0 ) k + 1 ≤ T ≤ k + 1 r + ( Ω 0 ) k + 1 , (29) where r − r_- r − r + r_+ r +
We use the differential Harnack inequality of Andrews [4 Theorem 5.6] , in the form recorded in [12 Theorem 13] . The estimate is written at fixed Gauss normal and applies directly to the speed E k E_k E k
Lemma 2.3. Let M t M_t M t 0 ≤ t < T 0\leq t<T 0 ≤ t < T (2) . Then
∂ t F ( z , t ) + k k + 1 F ( z , t ) t ≥ 0 ( z ∈ S n , 0 < t < T ) . (30) \partial_tF(z,t)+\frac{k}{k+1}\frac{F(z,t)}{t}\geq0\qquad(z\in\mathbb{S}^n,\ 0<t<T). \tag*{(30)} ∂ t F ( z , t ) + k + 1 k t F ( z , t ) ≥ 0 ( z ∈ S n , 0 < t < T ) . (30) Consequently,
F ( z , t 2 ) ≥ ( t 1 t 2 ) k / ( k + 1 ) F ( z , t 1 ) ( 0 < t 1 ≤ t 2 < T ) . (31) F(z,t_2)\geq\left(\frac{t_1}{t_2}\right)^{k/(k+1)}F(z,t_1)\qquad(0<t_1\leq t_2<T). \tag*{(31)} F ( z , t 2 ) ≥ ( t 2 t 1 ) k / ( k + 1 ) F ( z , t 1 ) ( 0 < t 1 ≤ t 2 < T ) . (31) Mean-width normalization
Let x ∗ x_* x ∗ T T T b ( t ) b(t) b ( t ) (3) is unchanged by translation. We use the normalized flow Y = ( X − x ∗ ) / b ( t ) Y=(X-x_*)/b(t) Y = ( X − x ∗ ) / b ( t )
u ( z , τ ( t ) ) = h ( z , t ) − ⟨ x ∗ , z ⟩ b ( t ) , (32) u(z,\tau(t))=\frac{h(z,t)-\langle x_*,z\rangle}{b(t)}, \tag*{(32)} u ( z , τ ( t )) = b ( t ) h ( z , t ) − ⟨ x ∗ , z ⟩ , (32) B ~ i j = ∇ ‾ i ∇ ‾ j u + u σ i j , W ~ = B ~ − 1 . (33) \widetilde{B}_{ij}=\overline{\nabla}_i\overline{\nabla}_j u+u\sigma_{ij},\qquad\widetilde{W}=\widetilde{B}^{-1}. \tag*{(33)} B ij = ∇ i ∇ j u + u σ ij , W = B − 1 . (33) The enclosed body is Ω ~ τ = ( Ω t − x ∗ ) / b ( t ) \widetilde{\Omega}_\tau=(\Omega_t-x_*)/b(t) Ω τ = ( Ω t − x ∗ ) / b ( t )
∫ S n u d σ = ∫ S n E 1 ( B ~ ) d σ = 1. (34) \int_{\mathbb{S}^n}u\,d\sigma=\int_{\mathbb{S}^n}E_1(\widetilde{B})\,d\sigma=1. \tag*{(34)} ∫ S n u d σ = ∫ S n E 1 ( B ) d σ = 1. (34) Thus the mean width is two. Equivalently, the mean perimeter of the planar projections is 2 π 2\pi 2 π
1 2 π ∫ Gr ( n + 1 , 2 ) Length ( ∂ ( Ω ~ τ ∣ Π ) ) d Π = ∫ S n u d σ . \frac{1}{2\pi}\int_{\operatorname{Gr}(n+1,2)}\operatorname{Length}(\partial(\widetilde{\Omega}_\tau\mid\Pi))\,d\Pi=\int_{\mathbb{S}^n}u\,d\sigma. 2 π 1 ∫ Gr ( n + 1 , 2 ) Length ( ∂ ( Ω τ ∣ Π )) d Π = ∫ S n u d σ . Here d Π d\Pi d Π
Set
F ‾ ( τ ) = ∫ S n E k ( B ~ − 1 ) d σ . (35) \overline{F}(\tau)=\int_{\mathbb{S}^n}E_k(\widetilde{B}^{-1})\,d\sigma. \tag*{(35)} F ( τ ) = ∫ S n E k ( B − 1 ) d σ . (35) Writing r ~ i \widetilde{r}_i r i B ~ \widetilde{B} B
B = b B ~ , W = b − 1 W ~ , r i = b r ~ i , F = b − k E k ( B ~ − 1 ) , B=b\widetilde{B},\qquad W=b^{-1}\widetilde{W},\qquad r_i=b\widetilde{r}_i,\qquad F=b^{-k}E_k(\widetilde{B}^{-1}), B = b B , W = b − 1 W , r i = b r i , F = b − k E k ( B − 1 ) , d t d τ = b k + 1 , d b d t = − b − k F ‾ , d b d τ = − F ‾ b . (36) \frac{dt}{d\tau}=b^{k+1},\qquad\frac{db}{dt}=-b^{-k}\overline{F},\qquad\frac{db}{d\tau}=-\overline{F}b. \tag*{(36)} d τ d t = b k + 1 , d t d b = − b − k F , d τ d b = − F b . (36) In the last identity b b b t ( τ ) t(\tau) t ( τ )
F ‾ ≥ ∫ S n E 1 ( B ~ ) − k d σ ≥ ( ∫ S n E 1 ( B ~ ) d σ ) − k = 1. (37) \overline{F}\geq\int_{\mathbb{S}^n}E_1(\widetilde{B})^{-k}\,d\sigma\geq\left(\int_{\mathbb{S}^n}E_1(\widetilde{B})\,d\sigma\right)^{-k}=1. \tag*{(37)} F ≥ ∫ S n E 1 ( B ) − k d σ ≥ ( ∫ S n E 1 ( B ) d σ ) − k = 1. (37) Lemma 5.1 will show that τ ( t ) → ∞ \tau(t) \to\infty τ ( t ) → ∞ t → T t \to T t → T
We retain the extinction-time scale only for sphere comparison. A sphere evolving by (2) satisfies
r ′ ( t ) = − r ( t ) − k , r ( t ) k + 1 = r ( 0 ) k + 1 − ( k + 1 ) t . (38) r'(t)=-r(t)^{-k}, \qquad r(t)^{k+1}=r(0)^{k+1}-(k+1)t. \tag*{(38)} r ′ ( t ) = − r ( t ) − k , r ( t ) k + 1 = r ( 0 ) k + 1 − ( k + 1 ) t . (38) Set
ℓ ( t ) = ( ( k + 1 ) ( T − t ) ) 1 / ( k + 1 ) . (39) \ell(t)=\left((k+1)(T-t)\right)^{1/(k+1)}. \tag*{(39)} ℓ ( t ) = ( ( k + 1 ) ( T − t ) ) 1/ ( k + 1 ) . (39) Comparison with spheres gives
r − ( Ω t ) ≤ ℓ ( t ) ≤ diam Ω t . (40) r_{-}(\Omega_t)\leq\ell(t)\leq\operatorname{diam}\Omega_t. \tag*{(40)} r − ( Ω t ) ≤ ℓ ( t ) ≤ diam Ω t . (40) Lemma 2.4. The normalized support function satisfies
∂ τ u = F ‾ u + G ( B ~ ) = F ‾ u − E k ( B ~ − 1 ) . (41) \partial_\tau u=\overline{F}u+G(\widetilde{B})=\overline{F}u-E_k(\widetilde{B}^{-1}). \tag*{(41)} ∂ τ u = F u + G ( B ) = F u − E k ( B − 1 ) . (41) Let C ~ G \widetilde{C}_G C G (26) evaluated at B ~ \widetilde{B} B
G ˙ i j ( B ) = b − ( k + 1 ) G ~ ˙ i j ( B ~ ) , C G = b − k C ~ G . (42) \dot{G}^{ij}(B)=b^{-(k+1)}\dot{\widetilde{G}}^{ij}(\widetilde{B}), \qquad C_G=b^{-k}\widetilde{C}_G. \tag*{(42)} G ˙ ij ( B ) = b − ( k + 1 ) G ˙ ij ( B ) , C G = b − k C G . (42) With the derivatives of G G G B ~ \widetilde{B} B
( ∂ τ − G ~ ˙ i j ∇ ‾ i ∇ ‾ j ) E k ( B ~ − 1 ) = ( G ~ ˙ i j σ i j − k F ‾ ) E k ( B ~ − 1 ) , (43) \left(\partial_\tau-\dot{\widetilde{G}}^{ij}\overline{\nabla}_i\overline{\nabla}_j\right)E_k(\widetilde{B}^{-1})=(\dot{\widetilde{G}}^{ij}\sigma_{ij}-k\overline{F})E_k(\widetilde{B}^{-1}), \tag*{(43)} ( ∂ τ − G ˙ ij ∇ i ∇ j ) E k ( B − 1 ) = ( G ˙ ij σ ij − k F ) E k ( B − 1 ) , (43) ( ∂ τ − G ~ ˙ i j ∇ ‾ i ∇ ‾ j ) E 1 ( B ~ ) = − 2 E 1 ( B ~ ) G ~ ˙ i j ∇ ‾ i E 1 ( B ~ ) ∇ ‾ j E 1 ( B ~ ) + ( k − 1 ) E k ( B ~ − 1 ) + ( F ‾ − G ~ ˙ i j σ i j ) E 1 ( B ~ ) − 1 n C ~ G , (44) \begin{aligned}
\left(\partial_\tau-\dot{\widetilde{G}}^{ij}\overline{\nabla}_i\overline{\nabla}_j\right)E_1(\widetilde{B})
={}&-\frac{2}{E_1(\widetilde{B})}\dot{\widetilde{G}}^{ij}\overline{\nabla}_iE_1(\widetilde{B})\overline{\nabla}_jE_1(\widetilde{B})+(k-1)E_k(\widetilde{B}^{-1})\\
&+(\overline{F}-\dot{\widetilde{G}}^{ij}\sigma_{ij})E_1(\widetilde{B})-\frac{1}{n}\widetilde{C}_G, \tag*{(44)}
\end{aligned} ( ∂ τ − G ˙ ij ∇ i ∇ j ) E 1 ( B ) = − E 1 ( B ) 2 G ˙ ij ∇ i E 1 ( B ) ∇ j E 1 ( B ) + ( k − 1 ) E k ( B − 1 ) + ( F − G ˙ ij σ ij ) E 1 ( B ) − n 1 C G , (44) and
( ∂ τ − G ~ ˙ i j ∇ ‾ i ∇ ‾ j ) ( E k ( B ~ − 1 ) E 1 ( B ~ ) ) = − 2 G ~ ˙ i j ∇ ‾ i log E 1 ( B ~ ) ∇ ‾ j ( E k ( B ~ − 1 ) E 1 ( B ~ ) ) + ( k − 1 ) E k ( B ~ − 1 ) 2 − ( k − 1 ) F ‾ E k ( B ~ − 1 ) E 1 ( B ~ ) − E k ( B ~ − 1 ) n C ~ G . (45) \begin{aligned}
\left(\partial_\tau-\dot{\widetilde{G}}^{ij}\overline{\nabla}_i\overline{\nabla}_j\right)\left(E_k(\widetilde{B}^{-1})E_1(\widetilde{B})\right)
={}&-2\dot{\widetilde{G}}^{ij}\overline{\nabla}_i\log E_1(\widetilde{B})\overline{\nabla}_j\left(E_k(\widetilde{B}^{-1})E_1(\widetilde{B})\right)\\
&+(k-1)E_k(\widetilde{B}^{-1})^2-(k-1)\overline{F}E_k(\widetilde{B}^{-1})E_1(\widetilde{B})\\
&-\frac{E_k(\widetilde{B}^{-1})}{n}\widetilde{C}_G. \tag*{(45)}
\end{aligned} ( ∂ τ − G ˙ ij ∇ i ∇ j ) ( E k ( B − 1 ) E 1 ( B ) ) = − 2 G ˙ ij ∇ i log E 1 ( B ) ∇ j ( E k ( B − 1 ) E 1 ( B ) ) + ( k − 1 ) E k ( B − 1 ) 2 − ( k − 1 ) F E k ( B − 1 ) E 1 ( B ) − n E k ( B − 1 ) C G . (45) Proof. The relations (36) give (41) . Homogeneity gives
G ~ ˙ i j ( b B ~ ) = b − ( k + 1 ) G ~ ˙ i j ( B ~ ) , G ~ ¨ i j , p q ( b B ~ ) = b − ( k + 2 ) G ~ ¨ i j , p q ( B ~ ) , \dot{\widetilde{G}}^{ij}(b\widetilde{B})=b^{-(k+1)}\dot{\widetilde{G}}^{ij}(\widetilde{B}), \qquad\ddot{\widetilde{G}}^{ij,pq}(b\widetilde{B})=b^{-(k+2)}\ddot{\widetilde{G}}^{ij,pq}(\widetilde{B}), G ˙ ij ( b B ) = b − ( k + 1 ) G ˙ ij ( B ) , G ¨ ij , pq ( b B ) = b − ( k + 2 ) G ¨ ij , pq ( B ) , and hence (42) . Rescaling (24) and (27) yields (43) and (44) . The product rule proves (45) . □
Lemma 2.5. Let 2 ≤ k ≤ n − 1 2\leq k\leq n-1 2 ≤ k ≤ n − 1 (41) satisfies
∥ u ( ⋅ , τ ) ∥ C 0 ≤ C , C − 1 σ ≤ B ~ ( ⋅ , τ ) ≤ C σ \|u(\cdot,\tau)\|_{C^0}\leq C,\qquad C^{-1}\sigma\leq\widetilde{B}(\cdot,\tau)\leq C\sigma ∥ u ( ⋅ , τ ) ∥ C 0 ≤ C , C − 1 σ ≤ B ( ⋅ , τ ) ≤ C σ for τ 0 ≤ τ < τ 1 \tau_0\leq\tau<\tau_1 τ 0 ≤ τ < τ 1 j , l j,l j , l C j , l = C j , l ( n , k , C , j , l ) C_{j,l}=C_{j,l}(n,k,C,j,l) C j , l = C j , l ( n , k , C , j , l )
∥ ∂ τ j u ( ⋅ , τ ) ∥ C l ( S n ) ≤ C j , l ( τ 0 + 1 ≤ τ < τ 1 ) . (46) \|\partial_\tau^j u(\cdot,\tau)\|_{C^l(\mathbb{S}^n)}\leq C_{j,l}\qquad(\tau_0+1\leq\tau<\tau_1). \tag*{(46)} ∥ ∂ τ j u ( ⋅ , τ ) ∥ C l ( S n ) ≤ C j , l ( τ 0 + 1 ≤ τ < τ 1 ) . (46) Proof. The radius bounds imply C ′ − 1 ≤ F ‾ ≤ C ′ C'^{-1}\leq\overline{F}\leq C' C ′ − 1 ≤ F ≤ C ′ [ a , a + 1 ] [a,a+1] [ a , a + 1 ]
λ ( τ ) = exp ( − ∫ a τ F ‾ ( ξ ) d ξ ) , v = λ u , d s d τ = λ k + 1 . \lambda(\tau)=\exp\left(-\int_a^\tau\overline{F}(\xi)\,d\xi\right),\qquad v=\lambda u,\qquad\frac{ds}{d\tau}=\lambda^{k+1}. λ ( τ ) = exp ( − ∫ a τ F ( ξ ) d ξ ) , v = λ u , d τ d s = λ k + 1 . Then
∂ s v = G ( ∇ ‾ 2 v + v σ ) . \partial_s v=G(\overline{\nabla}^2v+v\sigma). ∂ s v = G ( ∇ 2 v + v σ ) . @page 9/26 starts_mid=0 ends_mid=1
The spatial scaling and time change are uniformly bounded above and below on this interval. Since G G G [25 ] , followed by the parabolic Schauder estimates [26 Chapter IV] , gives uniform bounds for all spatial derivatives away from the initial time. These transfer to u u u (41) and (35) then gives the time derivative bounds. Applying the estimates on preceding intervals of length one proves (46) . □ \square □
Sharp Codazzi inequalities for E k E_k E k
The Codazzi identity (17) makes ∇ ‾ B \overline{\nabla} B ∇ B G ( B ) = − E k ( B − 1 ) G(B)=-E_k(B^{-1}) G ( B ) = − E k ( B − 1 ) [8 Sections 4–6] and [28 Lemma 2.1] . The additional gradient term below provides a quantitative lower bound for the rate of decrease of the monotone integral.
Proposition 3.1. Let n ≥ 2 n\ge2 n ≥ 2 1 ≤ k < n 1\le k<n 1 ≤ k < n G ( B ) = − E k ( B − 1 ) G(B)=-E_k(B^{-1}) G ( B ) = − E k ( B − 1 ) B > 0 B>0 B > 0 F = − G > 0 F=-G>0 F = − G > 0 T T T
− G ¨ i j , p q T m i j T m p q ≥ 2 n E 1 ( B ) G i j T i p p T j q q + 1 F ∑ m ( G i j T m i j ) 2 . (47) -\ddot{G}^{ij,pq}T_{mij}T_{mpq}\ge\frac{2}{nE_1(B)}G^{ij}T_{ipp}T_{jqq}+\frac{1}{F}\sum_m\left(G^{ij}T_{mij}\right)^2 . \tag*{(47)} − G ¨ ij , pq T mij T m pq ≥ n E 1 ( B ) 2 G ij T i pp T j q q + F 1 m ∑ ( G ij T mij ) 2 . (47) Moreover, the weaker inequality
− G ¨ i j , p q T m i j T m p q ≥ 2 n E 1 ( B ) G i j T i p p T j q q (48) -\ddot{G}^{ij,pq}T_{mij}T_{mpq}\ge\frac{2}{nE_1(B)}G^{ij}T_{ipp}T_{jqq} \tag*{(48)} − G ¨ ij , pq T mij T m pq ≥ n E 1 ( B ) 2 G ij T i pp T j q q (48) has equality if and only if T = 0 T=0 T = 0 K \mathcal{K} K (48) is bounded below by c ∣ T ∣ 2 c|T|^2 c ∣ T ∣ 2 c = c ( n , k , K ) > 0 c=c(n,k,\mathcal{K})>0 c = c ( n , k , K ) > 0 2 2 2 (48) is optimal.
Proof. It suffices to work with f ( B ) = σ k ( B − 1 ) f(B)=\sigma_k(B^{-1}) f ( B ) = σ k ( B − 1 ) s = tr B s=\operatorname{tr} B s = tr B v i = T i p p v_i=T_{ipp} v i = T i pp
Step 1. The case k = n − 1 k=n-1 k = n − 1 B i j = r i δ i j B_{ij}=r_i\delta_{ij} B ij = r i δ ij
f = s det B , log f = log s − log det B , f ˙ i i = f ( 1 s − 1 r i ) < 0 , f=\frac{s}{\det B},\qquad\log f=\log s-\log\det B,\qquad\dot{f}^{ii}=f\left(\frac{1}{s}-\frac{1}{r_i}\right)<0, f = det B s , log f = log s − log det B , f ˙ ii = f ( s 1 − r i 1 ) < 0 , where there is no summation in the last identity. Differentiating log f \log f log f
f ¨ i j , p q T m i j T m p q + 2 s f ˙ i j v i v j = 1 f ∑ m ( f ˙ i j T m i j ) 2 + f [ ∑ m , i , j T m i j 2 r i r j − ∑ i ( 2 s r i − 1 s 2 ) v i 2 ] . (49) \ddot{f}^{ij,pq}T_{mij}T_{mpq}+\frac{2}{s}\dot{f}^{ij}v_i v_j
=\frac{1}{f}\sum_m\left(\dot{f}^{ij}T_{mij}\right)^2
+f\left[\sum_{m,i,j}\frac{T_{mij}^2}{r_i r_j}-\sum_i\left(\frac{2}{sr_i}-\frac{1}{s^2}\right)v_i^2\right]. \tag*{(49)} f ¨ ij , pq T mij T m pq + s 2 f ˙ ij v i v j = f 1 m ∑ ( f ˙ ij T mij ) 2 + f [ m , i , j ∑ r i r j T mij 2 − i ∑ ( s r i 2 − s 2 1 ) v i 2 ] . (49) The full symmetry of T T T
∑ m , i , j T m i j 2 r i r j = ∑ i [ T i i i 2 r i 2 + ∑ j ≠ i ( 1 r j 2 + 2 r i r j ) T i j j 2 ] + 2 ∑ i < j < l ( 1 r i r j + 1 r i r l + 1 r j r l ) T i j l 2 . \sum_{m,i,j}\frac{T_{mij}^2}{r_i r_j}
=\sum_i\left[\frac{T_{iii}^2}{r_i^2}+\sum_{j\ne i}\left(\frac{1}{r_j^2}+\frac{2}{r_i r_j}\right)T_{ijj}^2\right]
+2\sum_{i<j<l}\left(\frac{1}{r_i r_j}+\frac{1}{r_i r_l}+\frac{1}{r_j r_l}\right)T_{ijl}^2. m , i , j ∑ r i r j T mij 2 = i ∑ r i 2 T iii 2 + j = i ∑ ( r j 2 1 + r i r j 2 ) T ij j 2 + 2 i < j < l ∑ ( r i r j 1 + r i r l 1 + r j r l 1 ) T ij l 2 . Fix i i i i i i
μ i = ∑ j ≠ i T i j j s − r i , v i = T i i i + ( s − r i ) μ i . \mu_i=\frac{\sum_{j\ne i}T_{ijj}}{s-r_i},\qquad v_i=T_{iii}+(s-r_i)\mu_i. μ i = s − r i ∑ j = i T ij j , v i = T iii + ( s − r i ) μ i . Since r j ≤ s − r i r_j \le s-r_i r j ≤ s − r i j ≠ i j \ne i j = i
∑ j ≠ i ( 1 r j 2 + 2 r i r j ) T i j j 2 ≥ 2 s − r i r i ( s − r i ) ∑ j ≠ i T i j j 2 r j ≥ 2 s − r i r i μ i 2 . \sum_{j\ne i}\left(\frac{1}{r_j^2}+\frac{2}{r_i r_j}\right)T_{ijj}^2 \ge\frac{2s-r_i}{r_i(s-r_i)}\sum_{j\ne i}\frac{T_{ijj}^2}{r_j}\ge\frac{2s-r_i}{r_i}\mu_i^2. j = i ∑ ( r j 2 1 + r i r j 2 ) T ij j 2 ≥ r i ( s − r i ) 2 s − r i j = i ∑ r j T ij j 2 ≥ r i 2 s − r i μ i 2 . Consequently,
T i i i 2 r i 2 + ∑ j ≠ i ( 1 r j 2 + 2 r i r j ) T i j j 2 ≥ T i i i 2 r i 2 + 2 s − r i r i μ i 2 = ( 2 s r i − 1 s 2 ) v i 2 + ( s − r i s r i T i i i − 2 s − r i s μ i ) 2 . (50) \begin{aligned}
\frac{T_{iii}^2}{r_i^2}+\sum_{j\ne i}\left(\frac{1}{r_j^2}+\frac{2}{r_i r_j}\right)T_{ijj}^2\ge\frac{T_{iii}^2}{r_i^2}+\frac{2s-r_i}{r_i}\mu_i^2 \\
&=\left(\frac{2}{sr_i}-\frac{1}{s^2}\right)v_i^2+\left(\frac{s-r_i}{sr_i}T_{iii}-\frac{2s-r_i}{s}\mu_i\right)^2. \tag*{(50)}
\end{aligned} r i 2 T iii 2 + j = i ∑ ( r j 2 1 + r i r j 2 ) T ij j 2 ≥ r i 2 T iii 2 + r i 2 s − r i μ i 2 = ( s r i 2 − s 2 1 ) v i 2 + ( s r i s − r i T iii − s 2 s − r i μ i ) 2 . (50) Summing over i i i (49) proves the strengthened inequality in this dimension.
If the weaker inequality is an equality, then f ˙ p q T i p q = 0 \dot f^{pq}T_{ipq}=0 f ˙ pq T i pq = 0 i i i (50) vanish, and equality holds in weighted Cauchy’s inequality. Thus T i j j / r j = μ i T_{ijj}/r_j=\mu_i T ij j / r j = μ i j ≠ i j\ne i j = i
∑ j T i j j r j − v i s = 0 , s − r i s r i T i i i − 2 s − r i s μ i = 0. \sum_j\frac{T_{ijj}}{r_j}-\frac{v_i}{s}=0,\qquad\frac{s-r_i}{sr_i}T_{iii}-\frac{2s-r_i}{s}\mu_i=0. j ∑ r j T ij j − s v i = 0 , s r i s − r i T iii − s 2 s − r i μ i = 0. Subtracting the second expression from the first gives μ i = 0 \mu_i=0 μ i = 0 T i i i = T i j j = 0 T_{iii}=T_{ijj}=0 T iii = T ij j = 0 i , j i,j i , j T = 0 T=0 T = 0
We will also use the following consequence of the same logarithmic differentiation. For every symmetric two-tensor V V V
f ¨ i j , p q V i j V p q − 1 f ( f ˙ i j V i j ) 2 = f [ ∑ i , j V i j 2 r i r j − 1 s 2 ( ∑ i V i i ) 2 ] ≥ f ( 1 − ∑ i r i 2 s 2 ) ∑ i , j V i j 2 r i r j . (51) \begin{aligned}
\ddot f^{ij,pq}V_{ij}V_{pq}-\frac{1}{f}\left(\dot f^{ij}V_{ij}\right)^2
=f\left[\sum_{i,j}\frac{V_{ij}^2}{r_i r_j}-\frac{1}{s^2}\left(\sum_i V_{ii}\right)^2\right] \\
&\ge f\left(1-\frac{\sum_i r_i^2}{s^2}\right)\sum_{i,j}\frac{V_{ij}^2}{r_i r_j}. \tag*{(51)}
\end{aligned} f ¨ ij , pq V ij V pq − f 1 ( f ˙ ij V ij ) 2 = f i , j ∑ r i r j V ij 2 − s 2 1 ( i ∑ V ii ) 2 ≥ f ( 1 − s 2 ∑ i r i 2 ) i , j ∑ r i r j V ij 2 . (51) The last step is Cauchy’s inequality, and the last expression is strictly positive for V ≠ 0 V\ne0 V = 0
Step 2. The general case. It remains to consider k < n − 1 k<n-1 k < n − 1 B B B ( k + 1 ) (k+1) ( k + 1 ) U ⊂ { 1 , … , n } U\subset\{1,\ldots,n\} U ⊂ { 1 , … , n } B U = ( B i j ) i , j ∈ U B_U=(B_{ij})_{i,j\in U} B U = ( B ij ) i , j ∈ U
s U = tr B U , f U = σ k ( B U − 1 ) = s U det B U . s_U=\operatorname{tr}B_U,\qquad f_U=\sigma_k(B_U^{-1})=\frac{s_U}{\det B_U}. s U = tr B U , f U = σ k ( B U − 1 ) = det B U s U . For any positive definite matrix B B B
σ k ( B − 1 ) = 1 n − k ∑ ∣ U ∣ = k + 1 σ k ( ( B − 1 ) U ) , \sigma_k(B^{-1})=\frac{1}{n-k}\sum_{|U|=k+1}\sigma_k\left((B^{-1})_U\right), σ k ( B − 1 ) = n − k 1 ∣ U ∣ = k + 1 ∑ σ k ( ( B − 1 ) U ) , since each principal minor of order k k k n − k n-k n − k
At a diagonal matrix, ( B − 1 ) U = ( B U ) − 1 (B^{-1})_U=(B_U)^{-1} ( B − 1 ) U = ( B U ) − 1 f U f_U f U B U B_U B U U U U
f ˙ U i i = f U ( 1 s U − 1 r i ) < 0 ( i ∈ U ) , \dot f_U^{ii}=f_U\left(\frac{1}{s_U}-\frac{1}{r_i}\right)<0\qquad(i\in U), f ˙ U ii = f U ( s U 1 − r i 1 ) < 0 ( i ∈ U ) , with no summation over i i i
f = 1 n − k ∑ U f U , f ˙ i j = 1 n − k ∑ U f ˙ U i j . (52) f=\frac{1}{n-k}\sum_U f_U,\qquad\dot f^{ij}=\frac{1}{n-k}\sum_U\dot f_U^{ij}. \tag*{(52)} f = n − k 1 U ∑ f U , f ˙ ij = n − k 1 U ∑ f ˙ U ij . (52) For the second variation, we must also account for the indices outside U U U V V V
[ ( B + ε V ) − 1 ] i j = δ i j r i − ε V i j r i r j + ε 2 ∑ l V i l V l j r i r l r j + O ( ε 3 ) . \left[(B+\varepsilon V)^{-1}\right]_{ij}=\frac{\delta_{ij}}{r_i}-\varepsilon\frac{V_{ij}}{r_ir_j}+\varepsilon^2\sum_l\frac{V_{il}V_{lj}}{r_ir_lr_j}+O(\varepsilon^3). [ ( B + ε V ) − 1 ] ij = r i δ ij − ε r i r j V ij + ε 2 l ∑ r i r l r j V i l V l j + O ( ε 3 ) . The corresponding expansion of ( B U + ε V U ) − 1 (B_U+\varepsilon V_U)^{-1} ( B U + ε V U ) − 1 l ∈ U l\in U l ∈ U
f ¨ i j , p q V i j V p q = 1 n − k ∑ U [ f ¨ U i j , p q V i j V p q − 2 ∑ i ∈ U ∑ j ∉ U f ˙ U i i r j V i j 2 ] . (53) \ddot{f}^{ij,pq}V_{ij}V_{pq}=\frac{1}{n-k}\sum_U\left[\ddot{f}^{ij,pq}_U V_{ij}V_{pq}-2\sum_{i\in U}\sum_{j\notin U}\frac{\dot{f}^{ii}_U}{r_j}V_{ij}^2\right]. \tag*{(53)} f ¨ ij , pq V ij V pq = n − k 1 U ∑ f ¨ U ij , pq V ij V pq − 2 i ∈ U ∑ j ∈ / U ∑ r j f ˙ U ii V ij 2 . (53) All sums over U U U ( k + 1 ) (k+1) ( k + 1 )
Apply this identity with V i j = T m i j V_{ij}=T_{mij} V ij = T mij m m m m ∈ U m\in U m ∈ U U U U m ∉ U m\notin U m ∈ / U (51) to ( T m i j ) i , j ∈ U (T_{mij})_{i,j\in U} ( T mij ) i , j ∈ U m = j m=j m = j T j j i = T i j j T_{jj i}=T_{ijj} T j j i = T ij j
f ¨ i j , p q T m i j T m p q ≥ 1 n − k ∑ U { − 2 ∑ i ∈ U f ¨ U i i [ ( ∑ j ∈ U T i j j ) 2 s U + ∑ j ∉ U T i j j 2 r j ] + 1 f U ∑ m ( f ¨ U i j T m i j ) 2 } . (54) \ddot{f}^{ij,pq}T_{mij}T_{mpq}\geq\frac{1}{n-k}\sum_U\left\{-2\sum_{i\in U}\ddot{f}^{ii}_U\left[\frac{\left(\sum_{j\in U}T_{ijj}\right)^2}{s_U}+\sum_{j\notin U}\frac{T_{ijj}^2}{r_j}\right]+\frac{1}{f_U}\sum_m\left(\ddot{f}^{ij}_U T_{mij}\right)^2\right\}. \tag*{(54)} f ¨ ij , pq T mij T m pq ≥ n − k 1 U ∑ ⎩ ⎨ ⎧ − 2 i ∈ U ∑ f ¨ U ii s U ( ∑ j ∈ U T ij j ) 2 + j ∈ / U ∑ r j T ij j 2 + f U 1 m ∑ ( f ¨ U ij T mij ) 2 ⎭ ⎬ ⎫ . (54) For each i ∈ U i\in U i ∈ U
( ∑ j ∈ U T i j j ) 2 s U + ∑ j ∉ U T i j j 2 r j ≥ ( ∑ j T i j j ) 2 s U + ∑ j ∉ U r j = v i 2 s . \frac{\left(\sum_{j\in U}T_{ijj}\right)^2}{s_U}+\sum_{j\notin U}\frac{T_{ijj}^2}{r_j}\geq\frac{\left(\sum_jT_{ijj}\right)^2}{s_U+\sum_{j\notin U}r_j}=\frac{v_i^2}{s}. s U ( ∑ j ∈ U T ij j ) 2 + j ∈ / U ∑ r j T ij j 2 ≥ s U + ∑ j ∈ / U r j ( ∑ j T ij j ) 2 = s v i 2 . Since f ¨ U i i < 0 \ddot{f}^{ii}_U<0 f ¨ U ii < 0 (52) gives
f ¨ i j , p q T m i j T m p q ≥ − 2 s f ¨ i j v i v j + 1 n − k ∑ U 1 f U ∑ m ( f ¨ U i j T m i j ) 2 . \ddot{f}^{ij,pq}T_{mij}T_{mpq}\geq-\frac{2}{s}\ddot{f}^{ij}v_iv_j+\frac{1}{n-k}\sum_U\frac{1}{f_U}\sum_m\left(\ddot{f}^{ij}_U T_{mij}\right)^2. f ¨ ij , pq T mij T m pq ≥ − s 2 f ¨ ij v i v j + n − k 1 U ∑ f U 1 m ∑ ( f ¨ U ij T mij ) 2 . For each fixed m m m (52) yields
1 n − k ∑ U ( f ¨ U i j T m i j ) 2 f U ≥ 1 f ( f ¨ i j T m i j ) 2 . \frac{1}{n-k}\sum_U\frac{\left(\ddot{f}^{ij}_U T_{mij}\right)^2}{f_U}\geq\frac{1}{f}\left(\ddot{f}^{ij}T_{mij}\right)^2. n − k 1 U ∑ f U ( f ¨ U ij T mij ) 2 ≥ f 1 ( f ¨ ij T mij ) 2 . This proves the strengthened inequality for f f f
For strictness of the weaker inequality, choose T a b c ≠ 0 T_{abc}\ne0 T ab c = 0 U U U b , c b,c b , c a ∈ U a\in U a ∈ U a ∉ U a\notin U a ∈ / U f ¨ U i j , p q T a i j T a p q \ddot{f}^{ij,pq}_U T_{aij}T_{apq} f ¨ U ij , pq T aij T a pq (51) and can be retained in addition to the trace terms when proving the weaker inequality. Thus the weaker inequality is strict for T ≠ 0 T\ne0 T = 0 ∣ T ∣ = 1 |T|=1 ∣ T ∣ = 1
Finally, take B = diag ( L , 1 , … , 1 ) B=\operatorname{diag}(L,1,\ldots,1) B = diag ( L , 1 , … , 1 ) T 111 = 1 T_{111}=1 T 111 = 1 s = L + n − 1 s=L+n-1 s = L + n − 1 v 1 = 1 v_1=1 v 1 = 1 v i = 0 v_i=0 v i = 0 i ≠ 1 i\ne1 i = 1
f ( B ) = ( n − 1 k ) + 1 L ( n − 1 k − 1 ) , f(B)=\binom{n-1}{k}+\frac{1}{L}\binom{n-1}{k-1}, f ( B ) = ( k n − 1 ) + L 1 ( k − 1 n − 1 ) , direct differentiation gives
f ¨ i j , p q T m i j T m p q − 1 f ∑ m ( f ˙ i j T m i j ) 2 − s − 1 f ˙ i j v i v j = L + n − 1 L ( 2 − k ( n − k ) L + k ) ⟶ 2 ( L ⟶ ∞ ) . \frac{\ddot f^{ij,pq}T_{mij}T_{mpq}-\frac{1}{f}\sum_m\left(\dot f^{ij}T_{mij}\right)^2}{-s^{-1}\dot f^{ij}v_i v_j}
=\frac{L+n-1}{L}\left(2-\frac{k}{(n-k)L+k}\right)\longrightarrow2\qquad(L\longrightarrow\infty). − s − 1 f ˙ ij v i v j f ¨ ij , pq T mij T m pq − f 1 ∑ m ( f ˙ ij T mij ) 2 = L L + n − 1 ( 2 − ( n − k ) L + k k ) ⟶ 2 ( L ⟶ ∞ ) . Here we used k < n k<n k < n (47) is optimal when the coefficient of the last term is fixed at 1. For the weaker inequality, the corresponding quotient is 2 ( L + n − 1 ) / L → 2 2(L+n-1)/L\to2 2 ( L + n − 1 ) / L → 2 (48) is also optimal.
Remark 3.1. The equality assertion in Proposition 3.1 concerns (48) , not the strengthened inequality (47) . For example, when n = 2 n=2 n = 2 k = 1 k=1 k = 1 B = Id B=\operatorname{Id} B = Id T 111 = 3 T_{111}=3 T 111 = 3 T 122 = T 212 = T 221 = 1 T_{122}=T_{212}=T_{221}=1 T 122 = T 212 = T 221 = 1 (47) .
Remark 3.2 (Relation to inverse-concavity). Set Φ ( B ) = log E k ( B − 1 ) \Phi(B)=\log E_k(B^{-1}) Φ ( B ) = log E k ( B − 1 ) log E k \log E_k log E k Φ \Phi Φ [8 ] , Theorem 2.1. In logarithmic form, (47) reads
− G ¨ i j , p q T m i j T m p q − 1 F ∑ m ( G ˙ i j T m i j ) 2 = F Φ ¨ i j , p q T m i j T m p q ≥ 2 n E 1 ( B ) G ˙ i j T i p p T j q q ≥ 0. -\ddot G^{ij,pq}T_{mij}T_{mpq}-\frac{1}{F}\sum_m\left(\dot G^{ij}T_{mij}\right)^2
=F\ddot\Phi^{ij,pq}T_{mij}T_{mpq}\ge\frac{2}{nE_1(B)}\dot G^{ij}T_{ipp}T_{jqq}\ge0. − G ¨ ij , pq T mij T m pq − F 1 m ∑ ( G ˙ ij T mij ) 2 = F Φ ¨ ij , pq T mij T m pq ≥ n E 1 ( B ) 2 G ˙ ij T i pp T j q q ≥ 0. Inverse-concavity gives the nonnegativity of the middle expression by summing over m m m T T T (47) is a quantitative Codazzi refinement of this traced inverse-concavity estimate.
Corollary 3.2. Let B = ∇ ‾ 2 h + h σ > 0 B=\overline{\nabla}^{2}h+h\sigma>0 B = ∇ 2 h + hσ > 0 1 ≤ k < n 1\le k<n 1 ≤ k < n (26) satisfies
C G ≥ ∣ ∇ ‾ E k ( B − 1 ) ∣ σ 2 E k ( B − 1 ) . (55) C_G\ge\frac{\left|\overline{\nabla}E_k(B^{-1})\right|_{\sigma}^{2}}{E_k(B^{-1})}. \tag*{(55)} C G ≥ E k ( B − 1 ) ∇ E k ( B − 1 ) σ 2 . (55) Moreover,
C G ≥ 0 , C G = 0 ⟺ ∇ ‾ B = 0 (56) C_G\ge0,\qquad C_G=0\Longleftrightarrow\overline{\nabla}B=0 \tag*{(56)} C G ≥ 0 , C G = 0 ⟺ ∇ B = 0 (56) at each point. For every compact subset K K K c = c ( n , k , K ) > 0 c=c(n,k,K)>0 c = c ( n , k , K ) > 0
C G ≥ c ∣ ∇ ‾ B ∣ 2 ( B ∈ K ) . (57) C_G\ge c\left|\overline{\nabla}B\right|^{2}\qquad(B\in K). \tag*{(57)} C G ≥ c ∇ B 2 ( B ∈ K ) . (57) The identity C G = 0 C_G=0 C G = 0 S n \mathbb{S}^{n} S n B ~ \widetilde B B C G ~ C_{\widetilde G} C G E k ( B − 1 ) E_k(B^{-1}) E k ( B − 1 ) E k ( B ~ − 1 ) E_k(\widetilde B^{-1}) E k ( B − 1 )
Proof. By the Codazzi identity, T m i j = ∇ ‾ m B i j T_{mij}=\overline{\nabla}_mB_{ij} T mij = ∇ m B ij
T m i i = n ∇ ‾ m E 1 ( B ) , G ˙ i j T m i j = − ∇ ‾ m E k ( B − 1 ) . T_{mii}=n\overline{\nabla}_mE_1(B),\qquad\dot G^{ij}T_{mij}=-\overline{\nabla}_mE_k(B^{-1}). T mii = n ∇ m E 1 ( B ) , G ˙ ij T mij = − ∇ m E k ( B − 1 ) . The assertions follow from Proposition 3.1. If C G ≡ 0 C_G\equiv0 C G ≡ 0 ∇ ‾ B = 0 \overline{\nabla}B=0 ∇ B = 0 σ \sigma σ B = r σ B=r\sigma B = r σ r > 0 r>0 r > 0
∇ ‾ 2 ( h − r ) + ( h − r ) σ = 0 , \overline{\nabla}^{2}(h-r)+(h-r)\sigma=0, ∇ 2 ( h − r ) + ( h − r ) σ = 0 , and hence h ( z ) = r + ⟨ c , z ⟩ h(z)=r+\langle c,z\rangle h ( z ) = r + ⟨ c , z ⟩ c ∈ R n + 1 c\in\mathbb{R}^{n+1} c ∈ R n + 1 (26) . The normalized assertions follow by applying the same argument to B ~ \widetilde B B
A monotone entropy and geometric estimates
We now assume n ≥ 3 n \ge3 n ≥ 3 2 ≤ k ≤ n − 1 2 \le k \le n-1 2 ≤ k ≤ n − 1 (6) :
p = n − 1 k − 1 , q = p − 1 = n − k k − 1 . p=\frac{n-1}{k-1},\qquad q=p-1=\frac{n-k}{k-1}. p = k − 1 n − 1 , q = p − 1 = k − 1 n − k . For a smooth closed strictly convex hypersurface M M M
E ( M ) = 1 ∣ S n ∣ ∫ M F p E 1 ( B ) q d μ = ∫ S n Z d σ , Z = E 1 ( B ) q E n − k ( B ) p ( det B ) q . (58) \mathcal{E}(M)=\frac{1}{\lvert S^n\rvert}\int_M F^p E_1(B)^q\,d\mu=\int_{S^n}Z\,d\sigma,\qquad Z=\frac{E_1(B)^qE_{n-k}(B)^p}{(\det B)^q}. \tag*{(58)} E ( M ) = ∣ S n ∣ 1 ∫ M F p E 1 ( B ) q d μ = ∫ S n Z d σ , Z = ( det B ) q E 1 ( B ) q E n − k ( B ) p . (58) Here F = E k ( W ) F=E_k(W) F = E k ( W )
In the following, we prove the monotonicity of E \mathcal{E} E E \mathcal{E} E (3) .
The entropy and its monotonicity
The integral (58) has a projection interpretation that explains its exponents. For background on integral-geometric formulas for convex bodies, see [30 Section 4.4] . Set j = n − k j=n-k j = n − k G r ( n + 1 , j + 1 ) \mathrm{Gr}(n+1,j+1) Gr ( n + 1 , j + 1 ) Π ∈ G r ( n + 1 , j + 1 ) \Pi\in\mathrm{Gr}(n+1,j+1) Π ∈ Gr ( n + 1 , j + 1 ) Ω Π = Ω ∣ Π \Omega_\Pi=\Omega|_\Pi Ω Π = Ω ∣ Π Ω \Omega Ω ν Π \nu_\Pi ν Π d μ Π d\mu_\Pi d μ Π Φ \Phi Φ S n S^n S n
∫ S n Φ ( z ) E j ( B ( z ) ) d σ ( z ) = 1 ∣ S j ∣ ∫ G r ( n + 1 , j + 1 ) ∫ ∂ Ω Π Φ ( ν Π ) d μ Π d Π . (59) \int_{S^n}\Phi(z)E_j(B(z))\,d\sigma(z)=\frac{1}{\lvert S^j\rvert}\int_{\mathrm{Gr}(n+1,j+1)}\int_{\partial\Omega_\Pi}\Phi(\nu_\Pi)\,d\mu_\Pi\,d\Pi. \tag*{(59)} ∫ S n Φ ( z ) E j ( B ( z )) d σ ( z ) = ∣ S j ∣ 1 ∫ Gr ( n + 1 , j + 1 ) ∫ ∂ Ω Π Φ ( ν Π ) d μ Π d Π. (59) Indeed, the support function of Ω Π \Omega_\Pi Ω Π h h h S ( Π ) = Π ∩ S n S(\Pi)=\Pi\cap S^n S ( Π ) = Π ∩ S n z ∈ S ( Π ) z\in S(\Pi) z ∈ S ( Π ) B ( z ) B(z) B ( z ) H = Π ∩ z ⊥ H=\Pi\cap z^\perp H = Π ∩ z ⊥ det ( B ( z ) ∣ H ) \det(B(z)|_H) det ( B ( z ) ∣ H )
∫ G r ( z ⊥ , j ) det ( B ( z ) ∣ H ) d H = E j ( B ( z ) ) , \int_{\mathrm{Gr}(z^\perp,j)}\det(B(z)|_H)\,dH=E_j(B(z)), ∫ Gr ( z ⊥ , j ) det ( B ( z ) ∣ H ) d H = E j ( B ( z )) , where d H dH d H ( z , H ) (z,H) ( z , H ) (59) .
Since F det B = E j ( B ) F\det B=E_j(B) F det B = E j ( B ) p = q + 1 p=q+1 p = q + 1
E ( M ) = 1 ∣ S j ∣ ∫ G r ( n + 1 , j + 1 ) ∫ ∂ Ω Π ( F E 1 ( B ) ) q ( ν Π ) d μ Π d Π . (60) \mathcal{E}(M)=\frac{1}{\lvert S^j\rvert}\int_{\mathrm{Gr}(n+1,j+1)}\int_{\partial\Omega_\Pi}(F E_1(B))^q(\nu_\Pi)\,d\mu_\Pi\,d\Pi. \tag*{(60)} E ( M ) = ∣ S j ∣ 1 ∫ Gr ( n + 1 , j + 1 ) ∫ ∂ Ω Π ( F E 1 ( B ) ) q ( ν Π ) d μ Π d Π. (60) Here F F F E 1 ( B ) E_1(B) E 1 ( B ) ν Π \nu_\Pi ν Π ∂ Ω Π \partial\Omega_\Pi ∂ Ω Π k = n − 1 k=n-1 k = n − 1 (60) is the average of their weighted boundary lengths, divided by 2 π 2\pi 2 π
The evolution equation (28) suggests considering the product F E 1 ( B ) F E_1(B) F E 1 ( B ) λ \lambda λ λ 1 − k \lambda^{1-k} λ 1 − k d μ Π d\mu_\Pi d μ Π λ j \lambda^j λ j (60) requires j − ( k − 1 ) q = 0 j-(k-1)q=0 j − ( k − 1 ) q = 0 q = ( n − k ) / ( k − 1 ) q=(n-k)/(k-1) q = ( n − k ) / ( k − 1 ) p = q + 1 p=q+1 p = q + 1
Proposition 4.1. (i). The entropy E \mathcal{E} E E ≥ 1 \mathcal{E} \ge1 E ≥ 1 E ( M ) = 1 \mathcal{E}(M)=1 E ( M ) = 1 M M M
(ii). Along a smooth closed strictly convex solution of (2) , write w = F E 1 ( B ) w=F E_1(B) w = F E 1 ( B )
d d t E ( M t ) = − ( n − k ) ∫ S n F 2 w q − 1 ( E 1 ( B ) E n − k − 1 ( B ) − E n − k ( B ) ) d σ \frac{d}{dt}\mathcal{E}(M_t)=-\left(n-k\right)\int_{\mathbb{S}^n}F^2w^{q-1}\left(E_1(B)E_{n-k-1}(B)-E_{n-k}(B)\right)\,d\sigma d t d E ( M t ) = − ( n − k ) ∫ S n F 2 w q − 1 ( E 1 ( B ) E n − k − 1 ( B ) − E n − k ( B ) ) d σ − p q ∫ S n w q − 2 E n − k ( B ) G ˙ i j ∇ ‾ i w ∇ ‾ j w d σ -pq\int_{\mathbb{S}^n}w^{q-2}E_{n-k}(B)\dot{G}^{ij}\overline{\nabla}_i w\overline{\nabla}_j w\,d\sigma − pq ∫ S n w q − 2 E n − k ( B ) G ˙ ij ∇ i w ∇ j w d σ − q n ∫ S n F w q − 1 E n − k ( B ) C G d σ ≤ 0. (61) -\frac{q}{n}\int_{\mathbb{S}^n}Fw^{q-1}E_{n-k}(B)C_G\,d\sigma\le0. \tag*{(61)} − n q ∫ S n F w q − 1 E n − k ( B ) C G d σ ≤ 0. (61) Equality holds at a time if and only if M t M_t M t
Proof. (i). Translation invariance is immediate, while dilation invariance follows from n − k p + q = 0 n-kp+q=0 n − k p + q = 0 ( n − k ) p = ( n − 1 ) q (n-k)p=(n-1)q ( n − k ) p = ( n − 1 ) q
Z = ( E 1 ( B ) ( det B ) 1 / n ) q ( E n − k ( B ) ( det B ) ( n − k ) / n ) p ≥ 1. Z=\left(\frac{E_1(B)}{(\det B)^{1/n}}\right)^q\left(\frac{E_{n-k}(B)}{(\det B)^{(n-k)/n}}\right)^p\ge1. Z = ( ( det B ) 1/ n E 1 ( B ) ) q ( ( det B ) ( n − k ) / n E n − k ( B ) ) p ≥ 1. Hence E ≥ 1 \mathcal{E}\ge1 E ≥ 1 E ( M ) = 1 \mathcal{E}(M)=1 E ( M ) = 1 Z = 1 Z=1 Z = 1 B = r σ B=r\sigma B = r σ r r r r r r M M M
(ii). Let N i j = ∂ E n − k ( B ) / ∂ B i j N^{ij}=\partial E_{n-k}(B)/\partial B_{ij} N ij = ∂ E n − k ( B ) / ∂ B ij (18) and ∂ t B = − ∇ ‾ 2 F − F σ \partial_tB=-\overline{\nabla}^2F-F\sigma ∂ t B = − ∇ 2 F − F σ
∂ t E n − k ( B ) = − ∇ ‾ i ( N i j ∇ ‾ j F ) − ( n − k ) F E n − k − 1 ( B ) . \partial_tE_{n-k}(B)=-\overline{\nabla}_i(N^{ij}\overline{\nabla}_jF)-(n-k)FE_{n-k-1}(B). ∂ t E n − k ( B ) = − ∇ i ( N ij ∇ j F ) − ( n − k ) F E n − k − 1 ( B ) . On the other hand, differentiating E n − k ( B ) = F det B E_{n-k}(B)=F\det B E n − k ( B ) = F det B
N i j = ( det B ) ( F ( B − 1 ) i j − G ˙ i j ) . N^{ij}=(\det B)\left(F(B^{-1})^{ij}-\dot{G}^{ij}\right). N ij = ( det B ) ( F ( B − 1 ) ij − G ˙ ij ) . Since both N i j N^{ij} N ij ( det B ) ( B − 1 ) i j (\det B)(B^{-1})^{ij} ( det B ) ( B − 1 ) ij
N i j ∇ ‾ i F = ∇ ‾ i ( E n − k ( B ) G ˙ i j ) − 2 ( det B ) G ˙ i j ∇ ‾ i F . (62) N^{ij}\overline{\nabla}_iF=\overline{\nabla}_i\left(E_{n-k}(B)\dot{G}^{ij}\right)-2(\det B)\dot{G}^{ij}\overline{\nabla}_iF. \tag*{(62)} N ij ∇ i F = ∇ i ( E n − k ( B ) G ˙ ij ) − 2 ( det B ) G ˙ ij ∇ i F . (62) We now differentiate E ( M t ) = ∫ S n w q E n − k ( B ) d σ \mathcal{E}(M_t)=\int_{\mathbb{S}^n}w^qE_{n-k}(B)\,d\sigma E ( M t ) = ∫ S n w q E n − k ( B ) d σ (28) , we obtain
d d t E ( M t ) = q ∫ S n w q − 1 E n − k ( B ) ( G ˙ i j ∇ ‾ i ∇ ‾ j w − 2 G ˙ i j ∇ ‾ i log E 1 ( B ) ∇ ‾ j w ) d σ − ∫ S n w q ∇ ‾ i ( N i j ∇ ‾ j F ) d σ + ∫ S n [ q ( k − 1 ) F 2 w q − 1 E n − k ( B ) − ( n − k ) F w q E n − k − 1 ( B ) ] d σ − q n ∫ S n F w q − 1 E n − k ( B ) C G d σ . \begin{aligned}
\frac{d}{dt}\mathcal{E}(M_t)
={}&q\int_{\mathbb{S}^n}w^{q-1}E_{n-k}(B)\left(\dot{G}^{ij}\overline{\nabla}_i\overline{\nabla}_jw-2\dot{G}^{ij}\overline{\nabla}_i\log E_1(B)\overline{\nabla}_jw\right)\,d\sigma\\
&-\int_{\mathbb{S}^n}w^q\overline{\nabla}_i(N^{ij}\overline{\nabla}_jF)\,d\sigma\\
&+\int_{\mathbb{S}^n}\left[q(k-1)F^2w^{q-1}E_{n-k}(B)-(n-k)Fw^qE_{n-k-1}(B)\right]\,d\sigma\\
&-\frac{q}{n}\int_{\mathbb{S}^n}Fw^{q-1}E_{n-k}(B)C_G\,d\sigma.
\end{aligned} d t d E ( M t ) = q ∫ S n w q − 1 E n − k ( B ) ( G ˙ ij ∇ i ∇ j w − 2 G ˙ ij ∇ i log E 1 ( B ) ∇ j w ) d σ − ∫ S n w q ∇ i ( N ij ∇ j F ) d σ + ∫ S n [ q ( k − 1 ) F 2 w q − 1 E n − k ( B ) − ( n − k ) F w q E n − k − 1 ( B ) ] d σ − n q ∫ S n F w q − 1 E n − k ( B ) C G d σ . Integrating by parts in the first two lines and using q ( k − 1 ) = ( n − k ) q(k-1)=(n-k) q ( k − 1 ) = ( n − k )
d d t E ( M t ) = − q ( q − 1 ) ∫ S n w q − 2 E n − k ( B ) G ˙ i j ∇ ‾ i w ∇ ‾ j w d σ − q ∫ S n w q − 1 [ ∇ ‾ i ( E n − k ( B ) G ˙ i j ) − N i j ∇ ‾ i F + 2 E n − k ( B ) G ˙ i j ∇ ‾ i log E 1 ( B ) ] ∇ ‾ j w d σ − ( n − k ) ∫ S n F 2 w q − 1 ( E 1 ( B ) E n − k − 1 ( B ) − E n − k ( B ) ) d σ − q n ∫ S n F w q − 1 E n − k ( B ) C G d σ . \begin{aligned}
\frac{d}{dt}\mathcal{E}(M_t)={}&-q(q-1)\int_{\mathbb{S}^n}w^{q-2}E_{n-k}(B)\dot{G}^{ij}\overline{\nabla}_i w\overline{\nabla}_j w\,d\sigma\\
&-q\int_{\mathbb{S}^n}w^{q-1}\left[\overline{\nabla}_i\left(E_{n-k}(B)\dot{G}^{ij}\right)-N^{ij}\overline{\nabla}_iF+2E_{n-k}(B)\dot{G}^{ij}\overline{\nabla}_i\log E_1(B)\right]\overline{\nabla}_j w\,d\sigma\\
&-(n-k)\int_{\mathbb{S}^n}F^2w^{q-1}\left(E_1(B)E_{n-k-1}(B)-E_{n-k}(B)\right)\,d\sigma\\
&-\frac{q}{n}\int_{\mathbb{S}^n}Fw^{q-1}E_{n-k}(B)C_G\,d\sigma.
\end{aligned} d t d E ( M t ) = − q ( q − 1 ) ∫ S n w q − 2 E n − k ( B ) G ˙ ij ∇ i w ∇ j w d σ − q ∫ S n w q − 1 [ ∇ i ( E n − k ( B ) G ˙ ij ) − N ij ∇ i F + 2 E n − k ( B ) G ˙ ij ∇ i log E 1 ( B ) ] ∇ j w d σ − ( n − k ) ∫ S n F 2 w q − 1 ( E 1 ( B ) E n − k − 1 ( B ) − E n − k ( B ) ) d σ − n q ∫ S n F w q − 1 E n − k ( B ) C G d σ . By (62) and w = F E 1 ( B ) w=FE_1(B) w = F E 1 ( B )
∇ ‾ i ( E n − k ( B ) G ˙ i j ) − N i j ∇ ‾ i F + 2 E n − k ( B ) G ˙ i j ∇ ‾ i log E 1 ( B ) = 2 ( det B ) G ˙ i j ( ∇ ‾ i F + F ∇ ‾ i log E 1 ( B ) ) = 2 w − 1 E n − k ( B ) G ˙ i j ∇ ‾ i w . \begin{aligned}
\overline{\nabla}_i\left(E_{n-k}(B)\dot{G}^{ij}\right)-N^{ij}\overline{\nabla}_iF+2E_{n-k}(B)\dot{G}^{ij}\overline{\nabla}_i\log E_1(B)
&=2(\det B)\dot{G}^{ij}\left(\overline{\nabla}_iF+F\overline{\nabla}_i\log E_1(B)\right)\\
&=2w^{-1}E_{n-k}(B)\dot{G}^{ij}\overline{\nabla}_i w.
\end{aligned} ∇ i ( E n − k ( B ) G ˙ ij ) − N ij ∇ i F + 2 E n − k ( B ) G ˙ ij ∇ i log E 1 ( B ) = 2 ( det B ) G ˙ ij ( ∇ i F + F ∇ i log E 1 ( B ) ) = 2 w − 1 E n − k ( B ) G ˙ ij ∇ i w . The two gradient contributions combine with coefficient − q ( q − 1 ) − 2 q = − q ( q + 1 ) = − p q -q(q-1)-2q=-q(q+1)=-pq − q ( q − 1 ) − 2 q = − q ( q + 1 ) = − pq p = q + 1 p=q+1 p = q + 1 (61) .
Finally, the Newton–Maclaurin inequality E 1 ( B ) E n − k − 1 ( B ) ≥ E n − k ( B ) E_1(B)E_{n-k-1}(B)\ge E_{n-k}(B) E 1 ( B ) E n − k − 1 ( B ) ≥ E n − k ( B ) ( G ˙ i j ) (\dot{G}^{ij}) ( G ˙ ij ) C G ≥ 0 C_G\ge0 C G ≥ 0 E ( M t ) \mathcal{E}(M_t) E ( M t ) C G ≡ 0 C_G\equiv0 C G ≡ 0 M t M_t M t
For 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 α = 1 \alpha=1 α = 1 [18 Theorem 1.4] .
Corollary 4.2. Let n ≥ 3 n\ge3 n ≥ 3 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 (2) is a family of round spheres.
Proof. Along a self-similar shrinking solution, the hypersurfaces differ only by translations and dilations. Thus E \mathcal{E} E
For the normalization (3) , write E ( τ ) = E ( Y ( τ ) ) \mathcal{E}(\tau)=\mathcal{E}(Y(\tau)) E ( τ ) = E ( Y ( τ ))
E ( τ ( t ) ) = E ( M t ) = ∫ S n E k ( B ~ − 1 ) p E 1 ( B ~ ) q det B ~ d σ . (63) \mathcal{E}(\tau(t))=\mathcal{E}(M_t)=\int_{\mathbb{S}^n}E_k(\widetilde{B}^{-1})^pE_1(\widetilde{B})^q\det\widetilde{B}\,d\sigma. \tag*{(63)} E ( τ ( t )) = E ( M t ) = ∫ S n E k ( B − 1 ) p E 1 ( B ) q det B d σ . (63) The density Z Z Z
Corollary 4.3. Along the normalized flow (3) , let
D ( τ ) = q n ∫ S n E k ( B ~ − 1 ) p E 1 ( B ~ ) q − 1 C ~ G det B ~ d σ . (64) \mathcal{D}(\tau)=\frac{q}{n}\int_{\mathbb{S}^n}E_k(\widetilde{B}^{-1})^pE_1(\widetilde{B})^{q-1}\widetilde{C}_G\det\widetilde{B}\,d\sigma. \tag*{(64)} D ( τ ) = n q ∫ S n E k ( B − 1 ) p E 1 ( B ) q − 1 C G det B d σ . (64) Then
d d τ E ( τ ) ≤ − D ( τ ) ≤ 0. (65) \frac{d}{d\tau}\mathcal{E}(\tau)\le-\mathcal{D}(\tau)\le0. \tag*{(65)} d τ d E ( τ ) ≤ − D ( τ ) ≤ 0. (65) In particular,
1 ≤ E ( τ ) ≤ E ( M 0 ) . (66) 1\le\mathcal{E}(\tau)\le\mathcal{E}(M_0). \tag*{(66)} 1 ≤ E ( τ ) ≤ E ( M 0 ) . (66) Proof. Since d t d τ = b k + 1 \frac{dt}{d\tau}=b^{k+1} d τ d t = b k + 1 d d τ E ( τ ) = b k + 1 d d t E ( M t ) \frac{d}{d\tau}\mathcal{E}(\tau)=b^{k+1}\frac{d}{dt}\mathcal{E}(M_t) d τ d E ( τ ) = b k + 1 d t d E ( M t ) (36) , (42) , and q ( k − 1 ) = n − k q(k-1)=n-k q ( k − 1 ) = n − k
b k + 1 F w q − 1 E n − k ( B ) C G = E k ( B ~ − 1 ) p E 1 ( B ~ ) q − 1 C ~ G det B ~ , b^{k+1}F w^{q-1}E_{n-k}(B)\mathcal{C}_{G}=E_k(\widetilde{B}^{-1})^p E_1(\widetilde{B})^{q-1}\widetilde{\mathcal{C}}_{G}\det\widetilde{B}, b k + 1 F w q − 1 E n − k ( B ) C G = E k ( B − 1 ) p E 1 ( B ) q − 1 C G det B , where p = q + 1 p=q+1 p = q + 1 (61) yields (65) . The bounds in (66) follow from Proposition 4.1. □
When k = n − 1 k=n-1 k = n − 1 (61) vanishes identically. The strict remainder in (64) is available throughout 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1
Inradius estimate
We aim to control the inradius in terms of E \mathcal{E} E
Lemma 4.4. For every B > 0 B>0 B > 0
E n − k ( B ) ≥ E 1 ( B ) k / ( n − 1 ) ( det B ) ( n − k − 1 ) / ( n − 1 ) . (67) E_{n-k}(B)\ge E_1(B)^{k/(n-1)}(\det B)^{(n-k-1)/(n-1)}. \tag*{(67)} E n − k ( B ) ≥ E 1 ( B ) k / ( n − 1 ) ( det B ) ( n − k − 1 ) / ( n − 1 ) . (67) Z ≥ ( E 1 ( B ) n det B ) 1 / ( k − 1 ) . (68) Z\ge\left(\frac{E_1(B)^n}{\det B}\right)^{1/(k-1)}. \tag*{(68)} Z ≥ ( det B E 1 ( B ) n ) 1/ ( k − 1 ) . (68) Proof. The Newton inequalities imply that log E j ( B ) \log E_j(B) log E j ( B ) j j j
log E n − k ( B ) ≥ k n − 1 log E 1 ( B ) + n − k − 1 n − 1 log det B , \log E_{n-k}(B)\ge\frac{k}{n-1}\log E_1(B)+\frac{n-k-1}{n-1}\log\det B, log E n − k ( B ) ≥ n − 1 k log E 1 ( B ) + n − 1 n − k − 1 log det B , which proves (67) . Therefore,
Z ≥ E 1 ( B ) q + k p / ( n − 1 ) ( det B ) − q + ( n − k − 1 ) p / ( n − 1 ) = ( E 1 ( B ) n det B ) 1 / ( k − 1 ) . Z\ge E_1(B)^{q+kp/(n-1)}(\det B)^{-q+(n-k-1)p/(n-1)}
=\left(\frac{E_1(B)^n}{\det B}\right)^{1/(k-1)}. Z ≥ E 1 ( B ) q + k p / ( n − 1 ) ( det B ) − q + ( n − k − 1 ) p / ( n − 1 ) = ( det B E 1 ( B ) n ) 1/ ( k − 1 ) . The ratio E 1 ( B ) / ( det B ) 1 / n E_1(B)/(\det B)^{1/n} E 1 ( B ) / ( det B ) 1/ n (68) gives
∫ S n [ ( E 1 ( B ) ( det B ) 1 / n ) n / ( k − 1 ) − 1 ] d σ ≤ E ( ∂ Ω ) − 1. \int_{\mathbb{S}^n}\left[\left(\frac{E_1(B)}{(\det B)^{1/n}}\right)^{n/(k-1)}-1\right]\,d\sigma\le\mathcal{E}(\partial\Omega)-1. ∫ S n [ ( ( det B ) 1/ n E 1 ( B ) ) n / ( k − 1 ) − 1 ] d σ ≤ E ( ∂ Ω ) − 1. Thus the entropy deficit controls, in an integral sense, the failure of the principal radii to be equal.
Proposition 4.5. Let n ≥ 3 n\ge3 n ≥ 3 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 Ω ⊂ R n + 1 \Omega\subset\mathbb{R}^{n+1} Ω ⊂ R n + 1 h h h B = ∇ ‾ 2 h + h σ B=\overline{\nabla}^{2}h+h\sigma B = ∇ 2 h + hσ
b = ∫ S n h d σ = ∫ S n E 1 ( B ) d σ . b=\int_{\mathbb{S}^n}h\,d\sigma=\int_{\mathbb{S}^n}E_1(B)\,d\sigma. b = ∫ S n h d σ = ∫ S n E 1 ( B ) d σ . Then
diam Ω ≤ C ( n ) b . (69) \operatorname{diam}\Omega\le C(n)b. \tag*{(69)} diam Ω ≤ C ( n ) b . (69) For each e ∈ S n e\in\mathbb{S}^n e ∈ S n
w e = h ( e ) + h ( − e ) , ψ ( z ) = 1 − ⟨ z , e ⟩ 2 . w_e=h(e)+h(-e),\qquad\psi(z)=\sqrt{1-\langle z,e\rangle^2}. w e = h ( e ) + h ( − e ) , ψ ( z ) = 1 − ⟨ z , e ⟩ 2 . We have
∫ S n det B ψ d σ ≤ C ( n ) w e b n − 1 , (70) \int_{\mathbb{S}^n}\det B\,\psi\,d\sigma\le C(n)w_e b^{n-1}, \tag*{(70)} ∫ S n det B ψ d σ ≤ C ( n ) w e b n − 1 , (70) and
min e ∈ S n w e ≤ ( n + 2 ) r − ( Ω ) . (71) \min_{e\in\mathbb{S}^n}w_e\le(n+2)r_-(\Omega). \tag*{(71)} e ∈ S n min w e ≤ ( n + 2 ) r − ( Ω ) . (71) Moreover, there is a constant c = c ( n , k ) > 0 c=c(n,k)>0 c = c ( n , k ) > 0
r − ( Ω ) ≥ c b E ( ∂ Ω ) − ( k − 1 ) . (72) r_{-}(\Omega)\ge cb\mathcal{E}(\partial\Omega)^{-(k-1)}. \tag*{(72)} r − ( Ω ) ≥ c b E ( ∂ Ω ) − ( k − 1 ) . (72) Proof. The equality defining b b b tr σ B = Δ ‾ h + n h \operatorname{tr}_{\sigma}B=\overline{\Delta}h+nh tr σ B = Δ h + nh b b b Ω \Omega Ω − 1 2 diam Ω e 1 -\frac{1}{2}\operatorname{diam}\Omega e_1 − 2 1 diam Ω e 1 1 2 diam Ω e 1 \frac{1}{2}\operatorname{diam}\Omega e_1 2 1 diam Ω e 1 h ( z ) ≥ 1 2 diam Ω ∣ z 1 ∣ h(z)\ge\frac{1}{2}\operatorname{diam}\Omega|z_1| h ( z ) ≥ 2 1 diam Ω∣ z 1 ∣
b ≥ 1 2 diam Ω ∫ S n ∣ z 1 ∣ d σ , b\ge\frac{1}{2}\operatorname{diam}\Omega\int_{\mathbb{S}^n}|z_1|\,d\sigma, b ≥ 2 1 diam Ω ∫ S n ∣ z 1 ∣ d σ , which proves (69) .
Fix e ∈ S n e\in\mathbb{S}^n e ∈ S n v ∈ e ⊥ v\in e^\perp v ∈ e ⊥ Ω ∣ v ⊥ \Omega|v^\perp Ω∣ v ⊥ w e w_e w e n − 1 n-1 n − 1 diam Ω \operatorname{diam}\Omega diam Ω [29 Section 2.4, formula (20)] yields
∫ ∂ Ω ∣ ⟨ ν , v ⟩ ∣ d μ = 2 Vol n ( Ω ∣ v ⊥ ) ≤ 2 w e ( diam Ω ) n − 1 . \int_{\partial\Omega}|\langle\nu,v\rangle|\,d\mu=2\operatorname{Vol}_n(\Omega|v^\perp)\le2w_e(\operatorname{diam}\Omega)^{n-1}. ∫ ∂ Ω ∣ ⟨ ν , v ⟩ ∣ d μ = 2 Vol n ( Ω∣ v ⊥ ) ≤ 2 w e ( diam Ω ) n − 1 . Averaging over v ∈ S n ∩ e ⊥ v\in\mathbb{S}^n\cap e^\perp v ∈ S n ∩ e ⊥ e ⊥ e^\perp e ⊥
∫ S n ∩ e ⊥ ∫ ∂ Ω ∣ ⟨ ν , v ⟩ ∣ d μ d v = ∫ ∂ Ω ( ∫ S n ∩ e ⊥ ∣ ⟨ ν , v ⟩ ∣ d v ) d μ = c n ∫ ∂ Ω 1 − ⟨ ν , e ⟩ 2 d μ = c n ∫ S n det B ψ d σ , \begin{aligned}
\int_{\mathbb{S}^n\cap e^\perp}\int_{\partial\Omega}|\langle\nu,v\rangle|\,d\mu\,dv
&=\int_{\partial\Omega}\left(\int_{\mathbb{S}^n\cap e^\perp}|\langle\nu,v\rangle|\,dv\right)d\mu\\
&=c_n\int_{\partial\Omega}\sqrt{1-\langle\nu,e\rangle^2}\,d\mu\\
&=c_n\int_{\mathbb{S}^n}\det B\,\psi\,d\sigma,
\end{aligned} ∫ S n ∩ e ⊥ ∫ ∂ Ω ∣ ⟨ ν , v ⟩ ∣ d μ d v = ∫ ∂ Ω ( ∫ S n ∩ e ⊥ ∣ ⟨ ν , v ⟩ ∣ d v ) d μ = c n ∫ ∂ Ω 1 − ⟨ ν , e ⟩ 2 d μ = c n ∫ S n det B ψ d σ , where c n > 0 c_n>0 c n > 0 n n n (69) gives
c n ∫ S n det B ψ d σ ≤ 2 w e ( diam Ω ) n − 1 ≤ C ( n ) w e b n − 1 . c_n\int_{\mathbb{S}^n}\det B\,\psi\,d\sigma\le2w_e(\operatorname{diam}\Omega)^{n-1}\le C(n)w_e b^{n-1}. c n ∫ S n det B ψ d σ ≤ 2 w e ( diam Ω ) n − 1 ≤ C ( n ) w e b n − 1 . This proves (70) .
The standard comparison between minimal width and inradius [10 (2.8)] implies (71) .
Finally, by (68) and Hölder’s inequality with exponents n / ( k − 1 ) n/(k-1) n / ( k − 1 ) n n n n / ( n − k ) n/(n-k) n / ( n − k )
b n ≤ C ( n , k ) E k − 1 ( ∫ S n det B ψ d σ ) ( ∫ S n ψ − 1 / ( n − k ) d σ ) n − k ≤ C ( n , k ) E k − 1 w e b n − 1 . b^n\le C(n,k)\mathcal{E}^{k-1}\left(\int_{\mathbb{S}^n}\det B\,\psi\,d\sigma\right)\left(\int_{\mathbb{S}^n}\psi^{-1/(n-k)}\,d\sigma\right)^{n-k}
\le C(n,k)\mathcal{E}^{k-1}w_e b^{n-1}. b n ≤ C ( n , k ) E k − 1 ( ∫ S n det B ψ d σ ) ( ∫ S n ψ − 1/ ( n − k ) d σ ) n − k ≤ C ( n , k ) E k − 1 w e b n − 1 . Here we used (70) . The integral involving ψ \psi ψ 1 / ( n − k ) ≤ 1 < n 1/(n-k)\le1<n 1/ ( n − k ) ≤ 1 < n
w e ≥ c ( n , k ) b E − ( k − 1 ) for every e ∈ S n . (73) w_e\ge c(n,k)b\mathcal{E}^{-(k-1)}\qquad\text{for every }e\in\mathbb{S}^n. \tag*{(73)} w e ≥ c ( n , k ) b E − ( k − 1 ) for every e ∈ S n . (73) The inradius estimate now follows from (71) . □
Quantitative stability of the entropy
The entropy also provides a quantitative measure of the deviation of a convex body from a round ball. The relevant geometric quantity is the mean area of its two-dimensional projections.
For a smooth strictly convex body Ω ⊂ R n + 1 \Omega\subset\mathbb{R}^{n+1} Ω ⊂ R n + 1 h h h
b = ∫ S n h d σ , c = ( n + 1 ) ∫ S n h ( z ) z d σ . b=\int_{\mathbb{S}^n}h\,d\sigma,\qquad c=(n+1)\int_{\mathbb{S}^n}h(z)z\,d\sigma. b = ∫ S n h d σ , c = ( n + 1 ) ∫ S n h ( z ) z d σ . Thus b b b c c c
A ( Ω ) = 1 π ∫ Gr ( n + 1 , 2 ) Area ( Ω ∣ Π ) d Π , \mathcal{A}(\Omega)=\frac{1}{\pi}\int_{\operatorname{Gr}(n+1,2)}\operatorname{Area}(\Omega\mathbin{|}\Pi)\,d\Pi, A ( Ω ) = π 1 ∫ Gr ( n + 1 , 2 ) Area ( Ω ∣ Π ) d Π , where d Π d\Pi d Π b b b A ( Ω ) = b 2 \mathcal{A}(\Omega)=b^2 A ( Ω ) = b 2
Proposition 4.6. Let n ≥ 3 n\geq3 n ≥ 3 2 ≤ k ≤ n − 1 2\leq k\leq n-1 2 ≤ k ≤ n − 1 Ω ⊂ R n + 1 \Omega\subset\mathbb{R}^{n+1} Ω ⊂ R n + 1
E ( ∂ Ω ) − ( k − 1 ) ≤ ∣ ∂ Ω ∣ ∣ S n ∣ b n ≤ 1. (74) \mathcal{E}(\partial\Omega)^{-(k-1)}\leq\frac{|\partial\Omega|}{|\mathbb{S}^n|b^n}\leq1. \tag*{(74)} E ( ∂ Ω ) − ( k − 1 ) ≤ ∣ S n ∣ b n ∣ ∂ Ω∣ ≤ 1. (74) and
E ( ∂ Ω ) − 2 ( k − 1 ) / n ≤ A ( Ω ) b 2 ≤ 1. (75) \mathcal{E}(\partial\Omega)^{-2(k-1)/n}\leq\frac{\mathcal{A}(\Omega)}{b^2}\leq1. \tag*{(75)} E ( ∂ Ω ) − 2 ( k − 1 ) / n ≤ b 2 A ( Ω ) ≤ 1. (75) Moreover,
d H ( Ω , B b ( c ) ‾ ) b ≤ C ( n , k ) ( E ( ∂ Ω ) − 1 ) 2 / ( n + 2 ) . (76) \frac{d_H(\Omega,\overline{B_b(c)})}{b}\leq C(n,k)\bigl(\mathcal{E}(\partial\Omega)-1\bigr)^{2/(n+2)}. \tag*{(76)} b d H ( Ω , B b ( c ) ) ≤ C ( n , k ) ( E ( ∂ Ω ) − 1 ) 2/ ( n + 2 ) . (76) In particular, E ( ∂ Ω ) = 1 \mathcal{E}(\partial\Omega)=1 E ( ∂ Ω ) = 1 Ω \Omega Ω
Proof. Write B = ∇ ‾ 2 h + h σ B=\overline{\nabla}^{2}h+h\sigma B = ∇ 2 h + hσ
A ( Ω ) = ∫ S n ( b 2 − 1 n ∣ ∇ ‾ h ∣ 2 ) d σ = ∫ S n h E 1 ( B ) d σ = ∫ S n E 2 ( B ) d σ . (77) \begin{aligned}
\mathcal{A}(\Omega)&=\int_{\mathbb{S}^n}\left(b^2-\frac{1}{n}|\overline{\nabla}h|^2\right)\,d\sigma\\
&=\int_{\mathbb{S}^n}hE_1(B)\,d\sigma=\int_{\mathbb{S}^n}E_2(B)\,d\sigma. \tag*{(77)}
\end{aligned} A ( Ω ) = ∫ S n ( b 2 − n 1 ∣ ∇ h ∣ 2 ) d σ = ∫ S n h E 1 ( B ) d σ = ∫ S n E 2 ( B ) d σ . (77) The last equality follows by integration by parts. The Alexandrov–Fenchel inequalities [30 Section 7.3] imply
( ∫ S n det B d σ ) 1 / n ≤ A ( Ω ) 1 / 2 ≤ b . \left(\int_{\mathbb{S}^n}\det B\,d\sigma\right)^{1/n}\leq\mathcal{A}(\Omega)^{1/2}\leq b. ( ∫ S n det B d σ ) 1/ n ≤ A ( Ω ) 1/2 ≤ b . On the other hand, (68) and Hölder’s inequality yield
b ≤ E ( ∂ Ω ) ( k − 1 ) / n ( ∫ S n det B d σ ) 1 / n . b\leq\mathcal{E}(\partial\Omega)^{(k-1)/n}\left(\int_{\mathbb{S}^n}\det B\,d\sigma\right)^{1/n}. b ≤ E ( ∂ Ω ) ( k − 1 ) / n ( ∫ S n det B d σ ) 1/ n . Combining these inequalities proves (74) and (75) .
By translation and dilation, we may assume c = 0 c=0 c = 0 b = 1 b=1 b = 1 h = 1 + v h=1+v h = 1 + v v v v (77) and the spectral gap of − Δ ‾ -\overline{\Delta} − Δ
∥ v ∥ H 1 ( S n ) 2 ≤ C ( n ) ( 1 − A ( Ω ) ) ≤ C ( n ) [ 1 − E ( ∂ Ω ) − 2 ( k − 1 ) / n ] . (78) \|v\|_{H^1(\mathbb{S}^n)}^2\leq C(n)(1-\mathcal{A}(\Omega))\leq C(n)\left[1-\mathcal{E}(\partial\Omega)^{-2(k-1)/n}\right]. \tag*{(78)} ∥ v ∥ H 1 ( S n ) 2 ≤ C ( n ) ( 1 − A ( Ω )) ≤ C ( n ) [ 1 − E ( ∂ Ω ) − 2 ( k − 1 ) / n ] . (78) All norms use the normalized spherical measure.
The inverse Gauss map and integration by parts give
∫ S n X d σ = ∫ S n ( h z + ∇ ‾ h ) d σ = c = 0 , \int_{\mathbb{S}^n}X\,d\sigma=\int_{\mathbb{S}^n}(hz+\overline{\nabla}h)\,d\sigma=c=0, ∫ S n X d σ = ∫ S n ( h z + ∇ h ) d σ = c = 0 , so 0 ∈ Ω 0\in\Omega 0 ∈ Ω R = max S n h R=\max_{\mathbb{S}^n}h R = max S n h a = ∥ h − 1 ∥ C 0 ( S n ) a=\|h-1\|_{C^0(\mathbb{S}^n)} a = ∥ h − 1 ∥ C 0 ( S n )
0 ≤ h ≤ R ≤ C ( n ) , Ω ⊂ B R ( 0 ) ‾ . 0\leq h\leq R\leq C(n),\qquad\Omega\subset\overline{B_R(0)}. 0 ≤ h ≤ R ≤ C ( n ) , Ω ⊂ B R ( 0 ) . Since X = h z + ∇ ‾ h X=hz+\overline{\nabla}h X = h z + ∇ h R − h ≤ 2 a R-h\le2a R − h ≤ 2 a
∣ ∇ ‾ h ∣ 2 = ∣ X ∣ 2 − h 2 ≤ ( R − h ) ( R + h ) ≤ 4 R a ≤ C ( n ) a . \begin{aligned}
\lvert\overline{\nabla}h\rvert^2=\lvert X\rvert^2-h^2 \\
&\le(R-h)(R+h)\le4Ra\le C(n)a.
\end{aligned} ∣ ∇ h ∣ 2 = ∣ X ∣ 2 − h 2 ≤ ( R − h ) ( R + h ) ≤ 4 R a ≤ C ( n ) a . If a > 0 a>0 a > 0 ∣ v ∣ \lvert v\rvert ∣ v ∣ ∣ v ∣ ≥ a / 2 \lvert v\rvert\ge a/2 ∣ v ∣ ≥ a /2 c ( n ) a c(n)\sqrt{a} c ( n ) a c ( n ) a n / 2 c(n)a^{n/2} c ( n ) a n /2 n ≥ 3 n\ge3 n ≥ 3
c ( n ) a ( n + 2 ) / 4 ≤ ∥ v ∥ L 2 n / ( n − 2 ) ( S n ) ≤ C ( n ) ∥ v ∥ H 1 ( S n ) . c(n)a^{(n+2)/4}\le\lVert v\rVert_{L^{2n/(n-2)}(\mathbb{S}^n)}\le C(n)\lVert v\rVert_{H^1(\mathbb{S}^n)}. c ( n ) a ( n + 2 ) /4 ≤ ∥ v ∥ L 2 n / ( n − 2 ) ( S n ) ≤ C ( n ) ∥ v ∥ H 1 ( S n ) . The case a = 0 a=0 a = 0 (78) and (75) , we obtain
∥ h − 1 ∥ C 0 ( S n ) ≤ C ( n ) ( 1 − A ( Ω ) ) 2 / ( n + 2 ) ≤ C ( n ) [ 1 − E ( ∂ Ω ) − 2 ( k − 1 ) / n ] 2 / ( n + 2 ) ≤ C ( n , k ) ( E ( ∂ Ω ) − 1 ) 2 / ( n + 2 ) . \begin{aligned}
\lVert h-1\rVert_{C^0(\mathbb{S}^n)}\le C(n)\left(1-\mathcal{A}(\Omega)\right)^{2/(n+2)} \\
&\le C(n)\left[1-\mathcal{E}(\partial\Omega)^{-2(k-1)/n}\right]^{2/(n+2)} \\
&\le C(n,k)\left(\mathcal{E}(\partial\Omega)-1\right)^{2/(n+2)}.
\end{aligned} ∥ h − 1 ∥ C 0 ( S n ) ≤ C ( n ) ( 1 − A ( Ω ) ) 2/ ( n + 2 ) ≤ C ( n ) [ 1 − E ( ∂ Ω ) − 2 ( k − 1 ) / n ] 2/ ( n + 2 ) ≤ C ( n , k ) ( E ( ∂ Ω ) − 1 ) 2/ ( n + 2 ) . Here we used 1 − s − β ≤ β ( s − 1 ) 1-s^{-\beta}\le\beta(s-1) 1 − s − β ≤ β ( s − 1 ) s ≥ 1 s\ge1 s ≥ 1 β > 0 \beta>0 β > 0 (76) , since the Hausdorff distance between convex bodies equals the uniform distance between their support functions. The equality statement follows as well. □
Remark 4.1. This proposition gives a static geometric interpretation of the entropy: its deficit controls the Hausdorff distance to the ball with the same mean width. We record this quantitative stability as an independent geometric consequence of the entropy. It will not be used in the subsequent proof of convergence of the normalized flow.
Smooth convergence to a round point
We now work with the normalized flow (3) , for which the mean of the support function is one. Unless otherwise stated, all constants depend only on M 0 M_0 M 0 n n n k k k
Lemma 5.1. Let 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 (3) is defined for all τ ≥ 0 \tau\ge0 τ ≥ 0
r − ( Ω ~ τ ) ≥ c , diam Y ( τ ) ≤ C , 0 < u ≤ C , (79) r_-(\widetilde{\Omega}_\tau)\ge c,\qquad\operatorname{diam}Y(\tau)\le C,\qquad0<u\le C, \tag*{(79)} r − ( Ω τ ) ≥ c , diam Y ( τ ) ≤ C , 0 < u ≤ C , (79) and
E k ( B ~ − 1 ) ≤ C , 1 ≤ F ‾ ≤ C , E k ( B ~ − 1 ) E 1 ( B ~ ) ≤ C . (80) E_k(\widetilde{B}^{-1})\le C,\qquad1\le\overline{F}\le C,\qquad E_k(\widetilde{B}^{-1})E_1(\widetilde{B})\le C. \tag*{(80)} E k ( B − 1 ) ≤ C , 1 ≤ F ≤ C , E k ( B − 1 ) E 1 ( B ) ≤ C . (80) Moreover,
c ℓ ( t ) ≤ b ( t ) ≤ C ℓ ( t ) ( 0 ≤ t < T ) . (81) c\ell(t)\le b(t)\le C\ell(t)\qquad(0\le t<T). \tag*{(81)} c ℓ ( t ) ≤ b ( t ) ≤ C ℓ ( t ) ( 0 ≤ t < T ) . (81) Proof. Since ∫ S n u d σ = 1 \int_{\mathbb{S}^n}u\,d\sigma=1 ∫ S n u d σ = 1 (79) . By strict nesting and extinction at x ∗ x_* x ∗ x ∗ ∈ int Ω t x_*\in\operatorname{int}\Omega_t x ∗ ∈ int Ω t t < T t<T t < T 0 < u ≤ diam Y ≤ C 0<u\le\operatorname{diam}Y\le C 0 < u ≤ diam Y ≤ C u u u x ∗ x_* x ∗
Scaling back and using (40) , we obtain
c b ( t ) ≤ r − ( Ω t ) ≤ ℓ ( t ) ≤ diam Ω t ≤ C b ( t ) , cb(t)\le r_-(\Omega_t)\le\ell(t)\le\operatorname{diam}\Omega_t\le Cb(t), c b ( t ) ≤ r − ( Ω t ) ≤ ℓ ( t ) ≤ diam Ω t ≤ C b ( t ) , which proves (81) . Since ℓ ( t ) k + 1 \ell(t)^{k+1} ℓ ( t ) k + 1 T − t T-t T − t
d τ d t = b ( t ) − ( k + 1 ) ≥ c T − t . \frac{d\tau}{dt}=b(t)^{-(k+1)}\ge\frac{c}{T-t}. d t d τ = b ( t ) − ( k + 1 ) ≥ T − t c . Hence τ ( t ) → ∞ \tau(t)\to\infty τ ( t ) → ∞ t → T t\to T t → T [23 Section 2] , we obtain
h x ∗ ( z , t ) = ∫ t T F ( z , s ) d s ≥ F ( z , t ) ∫ t T ( t s ) k / ( k + 1 ) d s ≥ ( t T ) k / ( k + 1 ) ( T − t ) F ( z , t ) , \begin{aligned}
h_{x_*}(z,t)=\int_t^T F(z,s)\,ds\ge F(z,t)\int_t^T\left(\frac{t}{s}\right)^{k/(k+1)}\,ds \\
&\ge\left(\frac{t}{T}\right)^{k/(k+1)}(T-t)F(z,t),
\end{aligned} h x ∗ ( z , t ) = ∫ t T F ( z , s ) d s ≥ F ( z , t ) ∫ t T ( s t ) k / ( k + 1 ) d s ≥ ( T t ) k / ( k + 1 ) ( T − t ) F ( z , t ) , where h x ∗ = h − ⟨ x ∗ , z ⟩ h_{x_*}=h-\langle x_*,z\rangle h x ∗ = h − ⟨ x ∗ , z ⟩
E k ( B ~ − 1 ) ≤ ( T t ) k / ( k + 1 ) b ( t ) k + 1 T − t u ≤ C (82) E_k(\widetilde B^{-1})\le\left(\frac{T}{t}\right)^{k/(k+1)}\frac{b(t)^{k+1}}{T-t}u\le C \tag*{(82)} E k ( B − 1 ) ≤ ( t T ) k / ( k + 1 ) T − t b ( t ) k + 1 u ≤ C (82) for T / 2 ≤ t < T T/2\le t<T T /2 ≤ t < T (81) and the uniform upper bound for u u u [ 0 , T / 2 ] [0,T/2] [ 0 , T /2 ] C C C (37) , this yields 1 ≤ F ‾ ≤ C 1\le\overline F\le C 1 ≤ F ≤ C
Recall that w = E k ( B ~ − 1 ) E 1 ( B ~ ) w=E_k(\widetilde B^{-1})E_1(\widetilde B) w = E k ( B − 1 ) E 1 ( B ) w w w (45) gives
∂ τ w ≤ ( k − 1 ) E k ( B ~ − 1 ) 2 − ( k − 1 ) F ‾ w ≤ ( k − 1 ) ( C − w ) . \partial_\tau w\le(k-1)E_k(\widetilde B^{-1})^2-(k-1)\overline Fw\le(k-1)(C-w). ∂ τ w ≤ ( k − 1 ) E k ( B − 1 ) 2 − ( k − 1 ) F w ≤ ( k − 1 ) ( C − w ) . The maximum principle therefore implies w ≤ C w\le C w ≤ C
Lemma 5.2. Let 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 u u u (32) . There is a constant C = C ( M 0 , n , k ) > 0 C=C(M_0,n,k)>0 C = C ( M 0 , n , k ) > 0 τ ≥ 0 \tau\ge0 τ ≥ 0
∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s , (83) \int_{\mathbb S^n}|u(z,\tau)-1|^2\,d\sigma\le C\int_\tau^\infty e^{-(s-\tau)\mathcal D(s)}\,ds, \tag*{(83)} ∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s , (83) where D \mathcal D D (64) . In particular,
u ( ⋅ , τ ) ⟶ 1 uniformly on S n as τ → ∞ . u(\cdot,\tau)\longrightarrow1\qquad\text{uniformly on }\mathbb S^n\quad\text{as }\tau\to\infty. u ( ⋅ , τ ) ⟶ 1 uniformly on S n as τ → ∞. Proof. In this proof, write F = E k ( B ~ − 1 ) F=E_k(\widetilde B^{-1}) F = E k ( B − 1 ) F ‾ = ∫ S n F d σ \overline F=\int_{\mathbb S^n}F\,d\sigma F = ∫ S n F d σ
0 < u ≤ C , 0 < F ≤ C , 1 ≤ F ‾ ≤ C , F E 1 ( B ~ ) ≤ C . 0<u\le C,\qquad0<F\le C,\qquad1\le\overline F\le C,\qquad FE_1(\widetilde B)\le C. 0 < u ≤ C , 0 < F ≤ C , 1 ≤ F ≤ C , F E 1 ( B ) ≤ C . Since Z = F p E 1 ( B ~ ) q det B ~ ≥ 1 Z=F^pE_1(\widetilde B)^q\det\widetilde B\ge1 Z = F p E 1 ( B ) q det B ≥ 1 (55) and the definition of D \mathcal D D
D ( τ ) ≥ q n ∫ S n Z F E 1 ( B ~ ) ∣ ∇ ‾ F ∣ 2 d σ ≥ c ∫ S n ∣ ∇ ‾ F ∣ 2 d σ . \mathcal D(\tau)\ge\frac{q}{n}\int_{\mathbb S^n}\frac{Z}{FE_1(\widetilde B)}|\overline\nabla F|^2\,d\sigma\ge c\int_{\mathbb S^n}|\overline\nabla F|^2\,d\sigma. D ( τ ) ≥ n q ∫ S n F E 1 ( B ) Z ∣ ∇ F ∣ 2 d σ ≥ c ∫ S n ∣ ∇ F ∣ 2 d σ . The scalar Poincaré inequality therefore yields
∫ S n ∣ F − F ‾ ∣ 2 d σ ≤ C D ( τ ) . \int_{\mathbb S^n}|F-\overline F|^2\,d\sigma\le C\mathcal D(\tau). ∫ S n ∣ F − F ∣ 2 d σ ≤ C D ( τ ) . By (41) ,
∂ τ ( u − 1 ) = F ‾ ( u − 1 ) − ( F − F ‾ ) . \partial_\tau(u-1)=\overline F(u-1)-(F-\overline F). ∂ τ ( u − 1 ) = F ( u − 1 ) − ( F − F ) . For S > τ S>\tau S > τ
u ( z , τ ) − 1 = e − ∫ τ S F ‾ ( ξ ) d ξ ( u ( z , S ) − 1 ) + ∫ τ S e − ∫ τ s F ‾ ( ξ ) d ξ ( F ( z , s ) − F ‾ ( s ) ) d s . u(z,\tau)-1=e^{-\int_\tau^S\overline F(\xi)\,d\xi}(u(z,S)-1)
+\int_\tau^S e^{-\int_\tau^s\overline F(\xi)\,d\xi}(F(z,s)-\overline F(s))\,ds. u ( z , τ ) − 1 = e − ∫ τ S F ( ξ ) d ξ ( u ( z , S ) − 1 ) + ∫ τ S e − ∫ τ s F ( ξ ) d ξ ( F ( z , s ) − F ( s )) d s . Since u u u F ‾ ≥ 1 \overline F\ge1 F ≥ 1 S → ∞ S\to\infty S → ∞ F F F F ‾ \overline F F
u ( z , τ ) − 1 = ∫ τ ∞ e − ∫ τ s F ‾ ( ξ ) d ξ ( F ( z , s ) − F ‾ ( s ) ) d s . u(z,\tau)-1=\int_\tau^\infty e^{-\int_\tau^s\overline F(\xi)\,d\xi}(F(z,s)-\overline F(s))\,ds. u ( z , τ ) − 1 = ∫ τ ∞ e − ∫ τ s F ( ξ ) d ξ ( F ( z , s ) − F ( s )) d s . Using F ‾ ≥ 1 \overline{F} \ge1 F ≥ 1
∫ τ ∞ e − ∫ τ s F ‾ ( ξ ) d ξ d s ≤ ∫ τ ∞ e − ( s − τ ) d s = 1. \int_{\tau}^{\infty} e^{-\int_{\tau}^{s}\overline{F}(\xi)\,d\xi}\,ds \le\int_{\tau}^{\infty} e^{-(s-\tau)}\,ds = 1. ∫ τ ∞ e − ∫ τ s F ( ξ ) d ξ d s ≤ ∫ τ ∞ e − ( s − τ ) d s = 1. The weighted Cauchy inequality and the speed estimate above give
∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ ∫ τ ∞ e − ∫ τ s F ‾ ( ξ ) d ξ ∫ S n ∣ F ( z , s ) − F ‾ ( s ) ∣ 2 d σ d s ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s . \begin{aligned}
\int_{\mathbb{S}^{n}} |u(z,\tau)-1|^{2}\,d\sigma\le\int_{\tau}^{\infty} e^{-\int_{\tau}^{s}\overline{F}(\xi)\,d\xi}\int_{\mathbb{S}^{n}} |F(z,s)-\overline{F}(s)|^{2}\,d\sigma\,ds \\
&\le C\int_{\tau}^{\infty}e^{-(s-\tau)}D(s)\,ds.
\end{aligned} ∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ ∫ τ ∞ e − ∫ τ s F ( ξ ) d ξ ∫ S n ∣ F ( z , s ) − F ( s ) ∣ 2 d σ d s ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s . This proves (83) .
By (65) and E ≥ 1 \mathcal{E}\ge1 E ≥ 1
∫ 0 ∞ D ( s ) d s ≤ E ( 0 ) − 1 < ∞ . \int_{0}^{\infty}D(s)\,ds\le\mathcal{E}(0)-1<\infty. ∫ 0 ∞ D ( s ) d s ≤ E ( 0 ) − 1 < ∞. Thus the right-hand side of (83) tends to zero, which proves the L 2 L^{2} L 2
Finally, since 0 ∈ Ω ~ τ 0\in\widetilde{\Omega}_{\tau} 0 ∈ Ω τ diam Ω ~ τ ≤ C \operatorname{diam}\widetilde{\Omega}_{\tau}\le C diam Ω τ ≤ C
∣ ∇ ‾ u ∣ 2 + u 2 = ∣ ∇ ‾ u + u z ∣ 2 ≤ C . |\overline{\nabla}u|^{2}+u^{2}=|\overline{\nabla}u+u z|^{2}\le C. ∣ ∇ u ∣ 2 + u 2 = ∣ ∇ u + u z ∣ 2 ≤ C . Integrating over a spherical cap about a maximum point of ∣ u − 1 ∣ |u-1| ∣ u − 1∣
∥ u ( ⋅ , τ ) − 1 ∥ C 0 ( S n ) n + 2 ≤ C ∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s ≤ C ∫ τ ∞ D ( s ) d s ⟶ 0. \begin{aligned}
\|u(\cdot,\tau)-1\|_{C^{0}(\mathbb{S}^{n})}^{n+2}\le C\int_{\mathbb{S}^{n}}|u(z,\tau)-1|^{2}\,d\sigma\\
&\le C\int_{\tau}^{\infty}e^{-(s-\tau)}D(s)\,ds \\
&\le C\int_{\tau}^{\infty}D(s)\,ds\longrightarrow0.
\end{aligned} ∥ u ( ⋅ , τ ) − 1 ∥ C 0 ( S n ) n + 2 ≤ C ∫ S n ∣ u ( z , τ ) − 1 ∣ 2 d σ ≤ C ∫ τ ∞ e − ( s − τ ) D ( s ) d s ≤ C ∫ τ ∞ D ( s ) d s ⟶ 0. This proves uniform convergence. □
We now prove Theorem 1.1 .
Proof of Theorem 1.1. Point extinction follows from Proposition 2.2 . We use (3) . Lemma 5.1 ensures that τ ∈ [ 0 , ∞ ) \tau\in[0,\infty) τ ∈ [ 0 , ∞ ) τ 1 \tau_{1} τ 1 u ≥ 1 / 2 u\ge1/2 u ≥ 1/2 τ ≥ τ 1 \tau\ge\tau_{1} τ ≥ τ 1 [10 ] , Theorem 5.2. For this step, write F = E k ( B ~ − 1 ) F=E_{k}(\widetilde{B}^{-1}) F = E k ( B − 1 ) G G G B ~ \widetilde{B} B (41) and (43) give
( ∂ τ − G ˙ i j ∇ ‾ i ∇ ‾ j ) u = ( F ‾ + G ˙ i j σ i j ) u − ( k + 1 ) F , (\partial_{\tau}-\dot{G}^{ij}\overline{\nabla}_{i}\overline{\nabla}_{j})u=(\overline{F}+\dot{G}^{ij}\sigma_{ij})u-(k+1)F, ( ∂ τ − G ˙ ij ∇ i ∇ j ) u = ( F + G ˙ ij σ ij ) u − ( k + 1 ) F , ( ∂ τ − G ˙ i j ∇ ‾ i ∇ ‾ j ) F = ( G ˙ i j σ i j − k F ‾ ) F . (\partial_{\tau}-\dot{G}^{ij}\overline{\nabla}_{i}\overline{\nabla}_{j})F=(\dot{G}^{ij}\sigma_{ij}-k\overline{F})F. ( ∂ τ − G ˙ ij ∇ i ∇ j ) F = ( G ˙ ij σ ij − k F ) F . The gradient terms in the logarithmic equations are nonnegative. Using F ‾ ≥ 1 \overline{F}\ge1 F ≥ 1
( ∂ τ − G ˙ i j ∇ ‾ i ∇ ‾ j ) log ( F u k + 1 ) ≥ F ‾ + ( k + 2 ) G ˙ i j σ i j − ( k + 1 ) 2 F u ≥ 1 − ( k + 1 ) 2 F u . (84) (\partial_{\tau}-\dot{G}^{ij}\overline{\nabla}_{i}\overline{\nabla}_{j})\log(Fu^{k+1})\ge\overline{F}+(k+2)\dot{G}^{ij}\sigma_{ij}-(k+1)^{2}\frac{F}{u}\ge1-(k+1)^{2}\frac{F}{u}. \tag*{(84)} ( ∂ τ − G ˙ ij ∇ i ∇ j ) log ( F u k + 1 ) ≥ F + ( k + 2 ) G ˙ ij σ ij − ( k + 1 ) 2 u F ≥ 1 − ( k + 1 ) 2 u F . (84) Let m ( τ ) = min S n F u k + 1 m(\tau)=\min_{\mathbb{S}^{n}}Fu^{k+1} m ( τ ) = min S n F u k + 1 F / u = m ( τ ) / u k + 2 ≤ 2 k + 2 m ( τ ) F/u=m(\tau)/u^{k+2}\le2^{k+2}m(\tau) F / u = m ( τ ) / u k + 2 ≤ 2 k + 2 m ( τ )
m ( τ ) ≥ min { m ( τ 1 ) , 1 2 k + 3 ( k + 1 ) 2 } > 0 ( τ ≥ τ 1 ) . m(\tau)\ge\min\left\{m(\tau_{1}),\frac{1}{2^{k+3}(k+1)^{2}}\right\}>0\qquad(\tau\ge\tau_{1}). m ( τ ) ≥ min { m ( τ 1 ) , 2 k + 3 ( k + 1 ) 2 1 } > 0 ( τ ≥ τ 1 ) . Since u ≤ C u\le C u ≤ C
E k ( B ~ − 1 ) ≥ c > 0 ( τ ≥ τ 1 ) . (85) E_{k}(\widetilde{B}^{-1})\ge c>0\qquad(\tau\ge\tau_{1}). \tag*{(85)} E k ( B − 1 ) ≥ c > 0 ( τ ≥ τ 1 ) . (85) The product bound in (80) now gives E 1 ( B ~ ) ≤ C E_1(\widetilde{B}) \le C E 1 ( B ) ≤ C κ i ( Y ) ≥ c \kappa_i(Y) \ge c κ i ( Y ) ≥ c
C ≥ E k ( B ~ − 1 ) ≥ k n c k − 1 κ max ( Y ) . C \ge E_k(\widetilde{B}^{-1}) \ge\frac{k}{n}c^{k-1}\kappa_{\max}(Y). C ≥ E k ( B − 1 ) ≥ n k c k − 1 κ m a x ( Y ) . Thus
c ≤ κ i ( Y ) ≤ C (86) c \le\kappa_i(Y) \le C \tag*{(86)} c ≤ κ i ( Y ) ≤ C (86) for all sufficiently large normalized times. Lemma 2.5 gives
∥ u ( ⋅ , τ ) ∥ C l ( S n ) ≤ C l ( l ≥ 0 ) . (87) \|u(\cdot,\tau)\|_{C^l(S^n)} \le C_l \qquad(l \ge0). \tag*{(87)} ∥ u ( ⋅ , τ ) ∥ C l ( S n ) ≤ C l ( l ≥ 0 ) . (87) Interpolation with the L 2 L^2 L 2
Finally, smooth convergence gives F ‾ ( τ ) → 1 \overline{F}(\tau) \to1 F ( τ ) → 1
d d t b ( t ) k + 1 = − ( k + 1 ) F ‾ ( τ ( t ) ) , b ( t ) ⟶ 0 ( t → T ) , \frac{d}{dt}b(t)^{k+1}=-(k+1)\overline{F}(\tau(t)),\qquad b(t)\longrightarrow0\quad(t\to T), d t d b ( t ) k + 1 = − ( k + 1 ) F ( τ ( t )) , b ( t ) ⟶ 0 ( t → T ) , integration yields
b ( t ) k + 1 ( k + 1 ) ( T − t ) ⟶ 1. (88) \frac{b(t)^{k+1}}{(k+1)(T-t)}\longrightarrow1. \tag*{(88)} ( k + 1 ) ( T − t ) b ( t ) k + 1 ⟶ 1. (88) Using κ i ( M t ) = b ( t ) − 1 κ i ( Y ( τ ( t ) ) ) \kappa_i(M_t)=b(t)^{-1}\kappa_i(Y(\tau(t))) κ i ( M t ) = b ( t ) − 1 κ i ( Y ( τ ( t ))) E k ( W ) = b ( t ) − k E k ( W ~ ) E_k(W)=b(t)^{-k}E_k(\widetilde{W}) E k ( W ) = b ( t ) − k E k ( W ) (5) from (86) . Increasing the upper constant covers the remaining compact interval of original time. □
We finally prove that the convergence in (4) is exponential in the normalized time.
Corollary 5.3. Let 2 ≤ k ≤ n − 1 2\le k\le n-1 2 ≤ k ≤ n − 1 u u u (3) . Set
μ ∗ = k − 1 + 2 k n . \mu_* = k-1+\frac{2k}{n}. μ ∗ = k − 1 + n 2 k . For every 0 < λ < μ ∗ 0<\lambda<\mu_* 0 < λ < μ ∗ j ≥ 0 j\ge0 j ≥ 0 C = C ( M 0 , n , k , j , λ ) > 0 C=C(M_0,n,k,j,\lambda)>0 C = C ( M 0 , n , k , j , λ ) > 0
∥ u ( ⋅ , τ ) − 1 ∥ C j ( S n ) ≤ C e − λ τ ( τ ≥ 0 ) . (89) \|u(\cdot,\tau)-1\|_{C^j(S^n)}\le Ce^{-\lambda\tau}\qquad(\tau\ge0). \tag*{(89)} ∥ u ( ⋅ , τ ) − 1 ∥ C j ( S n ) ≤ C e − λ τ ( τ ≥ 0 ) . (89) Proof. Write F = E k ( B ~ − 1 ) F=E_k(\widetilde{B}^{-1}) F = E k ( B − 1 ) F ‾ = ∫ S n F d σ \overline{F}=\int_{S^n}F\,d\sigma F = ∫ S n F d σ (41) and (34) ,
u τ = F ‾ u − F , ∫ S n u d σ = 1. u_\tau=\overline{F}u-F,\qquad\int_{S^n}u\,d\sigma=1. u τ = F u − F , ∫ S n u d σ = 1. The smooth convergence proved above gives u → 1 u\to1 u → 1 C ∞ ( S n ) C^\infty(S^n) C ∞ ( S n ) u u u
Let Π m \Pi_m Π m L 2 L^2 L 2 m m m
η = Π 1 ( u − 1 ) , w = ( I − Π 0 − Π 1 ) ( u − 1 ) . \eta=\Pi_1(u-1),\qquad w=(I-\Pi_0-\Pi_1)(u-1). η = Π 1 ( u − 1 ) , w = ( I − Π 0 − Π 1 ) ( u − 1 ) . The mean-width constraint gives u = 1 + η + w u=1+\eta+w u = 1 + η + w ∇ ‾ 2 η + η σ = 0 \overline{\nabla}^2\eta+\eta\sigma=0 ∇ 2 η + η σ = 0
B ~ = σ + Q , Q = ∇ ‾ 2 w + w σ . \widetilde{B}=\sigma+Q,\qquad Q=\overline{\nabla}^2w+w\sigma. B = σ + Q , Q = ∇ 2 w + w σ . Thus the curvature speed depends only on w w w
Recall that G ( B ) = − E k ( B − 1 ) G(B)=-E_k(B^{-1}) G ( B ) = − E k ( B − 1 ) G ˙ i j ( σ ) = k n σ i j \dot{G}^{ij}(\sigma)=\frac{k}{n}\sigma^{ij} G ˙ ij ( σ ) = n k σ ij
F = 1 − k n ( Δ ‾ + n ) w + R , F ‾ = 1 + R ‾ , (90) F=1-\frac{k}{n}(\overline{\Delta}+n)w+R,\qquad\overline{F}=1+\overline{R}, \tag*{(90)} F = 1 − n k ( Δ + n ) w + R , F = 1 + R , (90) where R ‾ = ∫ S n R d σ \overline{R}=\int_{S^n}R\,d\sigma R = ∫ S n R d σ
R = a i j Q i j , a i j = − ∫ 0 1 ( G ˙ i j ( σ + s Q ) − k n σ i j ) d s . R=a^{ij}Q_{ij},\qquad a^{ij}=-\int_0^1\left(\dot{G}^{ij}(\sigma+sQ)-\frac{k}{n}\sigma^{ij}\right)\,ds. R = a ij Q ij , a ij = − ∫ 0 1 ( G ˙ ij ( σ + s Q ) − n k σ ij ) d s . For all sufficiently large τ \tau τ G G G w → 0 w \to0 w → 0 C ∞ C^\infty C ∞
∥ a ∥ C 1 ( S n ) ≤ C ∥ w ∥ C 3 ( S n ) = o ( 1 ) , ∥ R ‾ ∥ C 0 ( S n ) ≤ C ∥ w ∥ C 2 ( S n ) 2 . (91) \|a\|_{C^1(S^n)} \le C\|w\|_{C^3(S^n)} = o(1), \qquad\|\overline{R}\|_{C^0(S^n)} \le C\|w\|_{C^2(S^n)}^2. \tag*{(91)} ∥ a ∥ C 1 ( S n ) ≤ C ∥ w ∥ C 3 ( S n ) = o ( 1 ) , ∥ R ∥ C 0 ( S n ) ≤ C ∥ w ∥ C 2 ( S n ) 2 . (91) Projecting the support equation onto harmonics of degree at least two, we obtain
w τ = k n Δ ‾ w + ( k + 1 ) w + R ‾ w − Π ≥ 2 R , Π ≥ 2 = I − Π 0 − Π 1 . w_\tau= \frac{k}{n}\overline{\Delta}w + (k+1)w + \overline{R}w - \Pi_{\geq2}R,\qquad\Pi_{\geq2}=I-\Pi_0-\Pi_1. w τ = n k Δ w + ( k + 1 ) w + R w − Π ≥ 2 R , Π ≥ 2 = I − Π 0 − Π 1 . Taking the L 2 L^2 L 2 w w w
1 2 d d τ ∥ w ∥ L 2 ( S n ) 2 = − k n ∥ ∇ ‾ w ∥ L 2 ( S n ) 2 + ( k + 1 + R ‾ ) ∥ w ∥ L 2 ( S n ) 2 − ∫ S n w R d σ . \frac{1}{2}\frac{d}{d\tau}\|w\|_{L^2(S^n)}^2
= -\frac{k}{n}\|\overline{\nabla}w\|_{L^2(S^n)}^2
+ (k+1+\overline{R})\|w\|_{L^2(S^n)}^2
-\int_{S^n}wR\,d\sigma. 2 1 d τ d ∥ w ∥ L 2 ( S n ) 2 = − n k ∥ ∇ w ∥ L 2 ( S n ) 2 + ( k + 1 + R ) ∥ w ∥ L 2 ( S n ) 2 − ∫ S n w R d σ . Integration by parts and (91) yield
∣ ∫ S n w R d σ ∣ ≤ C ∥ a ∥ C 1 ( S n ) ( ∥ ∇ ‾ w ∥ L 2 ( S n ) 2 + ∥ w ∥ L 2 ( S n ) 2 ) . \left|\int_{S^n}wR\,d\sigma\right|
\le C\|a\|_{C^1(S^n)}
\left(\|\overline{\nabla}w\|_{L^2(S^n)}^2+\|w\|_{L^2(S^n)}^2\right). ∫ S n w R d σ ≤ C ∥ a ∥ C 1 ( S n ) ( ∥ ∇ w ∥ L 2 ( S n ) 2 + ∥ w ∥ L 2 ( S n ) 2 ) . Since R ‾ = o ( 1 ) \overline{R}=o(1) R = o ( 1 ) ϵ > 0 \epsilon>0 ϵ > 0 τ \tau τ
1 2 d d τ ∥ w ∥ L 2 ( S n ) 2 ≤ − ( k n − ϵ ) ∥ ∇ ‾ w ∥ L 2 ( S n ) 2 + ( k + 1 + ϵ ) ∥ w ∥ L 2 ( S n ) 2 . \frac{1}{2}\frac{d}{d\tau}\|w\|_{L^2(S^n)}^2
\le-\left(\frac{k}{n}-\epsilon\right)\|\overline{\nabla}w\|_{L^2(S^n)}^2
+(k+1+\epsilon)\|w\|_{L^2(S^n)}^2. 2 1 d τ d ∥ w ∥ L 2 ( S n ) 2 ≤ − ( n k − ϵ ) ∥ ∇ w ∥ L 2 ( S n ) 2 + ( k + 1 + ϵ ) ∥ w ∥ L 2 ( S n ) 2 . The first eigenvalue of − Δ ‾ -\overline{\Delta} − Δ 2 ( n + 1 ) 2(n+1) 2 ( n + 1 )
∥ ∇ ‾ w ∥ L 2 ( S n ) 2 ≥ 2 ( n + 1 ) ∥ w ∥ L 2 ( S n ) 2 , \|\overline{\nabla}w\|_{L^2(S^n)}^2
\ge2(n+1)\|w\|_{L^2(S^n)}^2, ∥ ∇ w ∥ L 2 ( S n ) 2 ≥ 2 ( n + 1 ) ∥ w ∥ L 2 ( S n ) 2 , and consequently
1 2 d d τ ∥ w ∥ L 2 ( S n ) 2 ≤ − ( μ ∗ − ( 2 n + 3 ) ϵ ) ∥ w ∥ L 2 ( S n ) 2 . \frac{1}{2}\frac{d}{d\tau}\|w\|_{L^2(S^n)}^2
\le-\left(\mu_*-(2n+3)\epsilon\right)\|w\|_{L^2(S^n)}^2. 2 1 d τ d ∥ w ∥ L 2 ( S n ) 2 ≤ − ( μ ∗ − ( 2 n + 3 ) ϵ ) ∥ w ∥ L 2 ( S n ) 2 . It follows that, for every 0 < ν < μ ∗ 0<\nu<\mu_* 0 < ν < μ ∗
∥ w ( ⋅ , τ ) ∥ L 2 ( S n ) ≤ C ν e − ν τ . (92) \|w(\cdot,\tau)\|_{L^2(S^n)}\le C_\nu e^{-\nu\tau}. \tag*{(92)} ∥ w ( ⋅ , τ ) ∥ L 2 ( S n ) ≤ C ν e − ν τ . (92) Fix 0 < λ < μ ∗ 0<\lambda<\mu_* 0 < λ < μ ∗ λ < ν < μ ∗ \lambda<\nu<\mu_* λ < ν < μ ∗ j ≥ 0 j\ge0 j ≥ 0 N > s > j + n / 2 N>s>j+n/2 N > s > j + n /2 ν ( 1 − s / N ) > λ \nu(1-s/N)>\lambda ν ( 1 − s / N ) > λ H N H^N H N
∥ w ∥ C j ( S n ) ≤ C ∥ w ∥ H s ( S n ) ≤ C ∥ w ∥ L 2 ( S n ) 1 − s / N ∥ w ∥ H N ( S n ) s / N ≤ C j , λ e − λ τ . (93) \|w\|_{C^j(S^n)}
\le C\|w\|_{H^s(S^n)}
\le C\|w\|_{L^2(S^n)}^{1-s/N}\|w\|_{H^N(S^n)}^{s/N}
\le C_{j,\lambda}e^{-\lambda\tau}. \tag*{(93)} ∥ w ∥ C j ( S n ) ≤ C ∥ w ∥ H s ( S n ) ≤ C ∥ w ∥ L 2 ( S n ) 1 − s / N ∥ w ∥ H N ( S n ) s / N ≤ C j , λ e − λ τ . (93) It remains to control the first harmonics. Projecting (41) and using (90) , we obtain
η τ = F ‾ η − Π 1 R . \eta_\tau=\overline{F}\eta-\Pi_1R. η τ = F η − Π 1 R . For S > τ S>\tau S > τ
η ( τ ) = e − ∫ τ S F ‾ ( ξ ) d ξ η ( S ) + ∫ τ S e − ∫ τ s F ‾ ( ξ ) d ξ Π 1 R ( s ) d s . \eta(\tau)=e^{-\int_\tau^S\overline{F}(\xi)\,d\xi}\eta(S)
+\int_\tau^S e^{-\int_\tau^s\overline{F}(\xi)\,d\xi}\Pi_1R(s)\,ds. η ( τ ) = e − ∫ τ S F ( ξ ) d ξ η ( S ) + ∫ τ S e − ∫ τ s F ( ξ ) d ξ Π 1 R ( s ) d s . Since F ‾ ≥ 1 \overline{F}\ge1 F ≥ 1 (37) and η \eta η S → ∞ S\to\infty S → ∞
η ( τ ) = ∫ τ ∞ e − ∫ τ s F ‾ ( ξ ) d ξ Π 1 R ( s ) d s . \eta(\tau)=\int_\tau^\infty e^{-\int_\tau^s\overline{F}(\xi)\,d\xi}\Pi_1R(s)\,ds. η ( τ ) = ∫ τ ∞ e − ∫ τ s F ( ξ ) d ξ Π 1 R ( s ) d s . The first-harmonic space is finite dimensional. By (91) and (93) ,
∥ Π 1 R ( s ) ∥ C j ( S n ) ≤ C j ∥ R ( s ) ∥ C 0 ( S n ) ≤ C j ∥ w ( s ) ∥ C 2 ( S n ) 2 ≤ C j , λ e − 2 λ s . \|\Pi_1R(s)\|_{C^j(S^n)}
\le C_j\|R(s)\|_{C^0(S^n)}
\le C_j\|w(s)\|_{C^2(S^n)}^2
\le C_{j,\lambda}e^{-2\lambda s}. ∥ Π 1 R ( s ) ∥ C j ( S n ) ≤ C j ∥ R ( s ) ∥ C 0 ( S n ) ≤ C j ∥ w ( s ) ∥ C 2 ( S n ) 2 ≤ C j , λ e − 2 λ s . It follows that
∥ η ( τ ) ∥ C j ( S n ) ≤ C j , λ ∫ τ ∞ e − ( s − τ ) e − 2 λ s d s ≤ C j , λ e − 2 λ τ . \|\eta(\tau)\|_{C^j(S^n)}
\le C_{j,\lambda}\int_\tau^\infty e^{-(s-\tau)}e^{-2\lambda s}\,ds
\le C_{j,\lambda}e^{-2\lambda\tau}. ∥ η ( τ ) ∥ C j ( S n ) ≤ C j , λ ∫ τ ∞ e − ( s − τ ) e − 2 λ s d s ≤ C j , λ e − 2 λ τ . Combining this with (93) proves (89) for all sufficiently large τ \tau τ □ \square □
Acknowledgements
This work was supported by the National Key Research and Development Program of China (2021YFA1001800), the National Natural Science Foundation of China (Grant No. 12531002), and the Fundamental Research Funds for the Central Universities. The third author was also supported by the National Natural Science Foundation of China (Grant No. 12601099). The last author was also supported by the China Postdoctoral Science Foundation (Grant No. 2025M783146).
References
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
