Introduction
In this paper, we prove a uniform L 1 L^{1} L 1 [36 ] , as stated in [27 Conjecture 2.18] .
Theorem 1.1. Let ( M n , g , J , p ) (M^{n},g,J,p) ( M n , g , J , p ) n = 2 m ≥ 2 n=2m\geq2 n = 2 m ≥ 2 Ric g ≥ − ( n − 1 ) g \operatorname{Ric}_{g}\geq-(n-1)g Ric g ≥ − ( n − 1 ) g Vol g ( B 1 ( p ) ) ≥ v > 0 \operatorname{Vol}_{g}(B_{1}(p))\geq v>0 Vol g ( B 1 ( p )) ≥ v > 0
∫ B 1 ( p ) ∣ Ric g ∣ d V g ≤ C ( n , v ) . (1) \int_{B_{1}(p)} |\operatorname{Ric}_{g}| \, \mathrm{d}V_{g} \leq C(n,v). \tag*{(1)} ∫ B 1 ( p ) ∣ Ric g ∣ d V g ≤ C ( n , v ) . (1) Under a sectional-curvature lower bound, a local curvature integral estimate was established by Petrunin [31 ] , with an alternative proof given by Li [20 ] . The problem is more difficult under only a Ricci lower bound. The counterexamples of Hao and Zhu [14 ] and Xu [34 ] disprove the asymptotic form of Yau’s conjecture, while Xu’s collapsing examples also show that a uniform local bound cannot hold on general Riemannian manifolds without a noncollapsing assumption. Under a noncollapsing assumption, Jiang and Naber obtained L p L^{p} L p 0 < p < 1 0<p<1 0 < p < 1 [19 ] ; see also Naber [27 ] . A proof was later given by Cucinotta and Mondino [11 ] . Here we study the case p = 1 p=1 p = 1 [10 , 13 , 25 , 26 , 32 , 33 , 35 , 37 , 38 ] .
Kähler geometry provides additional structure for proving the L 1 L^{1} L 1 [9 ] , Ni [30 ] , and Liu [22 ] relates volume growth to scalar-curvature decay and integral estimates. Related developments include Liu’s Cohn–Vossen-type inequality for certain noncompact Kähler manifolds [21 ] and the Ricci-form estimates of Liu and Ni [23 ] under additional nonnegative-curvature and Ricci-pinching assumptions.
For the local problem under a Ricci lower bound, the results of Liu and Székelyhidi [24 ] are particularly relevant. On almost Euclidean balls, they constructed holomorphic coordinates and obtained local scalar-curvature integral estimates without any polarization assumption. For general noncollapsed unit balls, they proved the corresponding integral bound in the polarized setting. Under a Ricci lower bound, these scalar-curvature integral bounds are equivalent to local L 1 L^{1} L 1 Theorem 1.1 removes the polarization assumption from the estimate on general balls.
Our proof of this extension combines the almost Euclidean estimates with an analysis of regions close to singular cones. The foundational work of Cheeger and Colding [1 , 2 , 3 , 4 ] established the almost splitting theorem and the structure theory of noncollapsed limit spaces, providing the basic analytic and geometric framework for this approach. In the Kähler setting, the fundamental work of Cheeger, Colding, and Tian [5 ] showed how the complex structure gives stronger control of the singularities of limit spaces. In particular, they used the parallel complex structure to obtain almost holomorphic functions from real splitting functions. This idea also appears in [24 ] and is used in the complex pairing argument in Lemma 2.16 . Quantitative control of the singular geometry was developed by Cheeger and Naber [7 ] , who introduced quantitative stratification and obtained volume estimates for quantitative singular strata. For noncollapsed manifolds with bounded Ricci curvature, they also proved codimension four regularity and L q L^{q} L q q < 2 q<2 q < 2 [8 ] .
Building on this work, Jiang and Naber [18 ] proved the L 2 L^2 L 2 L 2 L^2 L 2 L 2 L^2 L 2 [15 , 16 , 17 , 28 , 29 ] .
Our proof follows this strategy in the codimension two setting, using the neck structure, decomposition, and entropy estimates of Cheeger, Jiang, and Naber [6 ] . The local analysis is more difficult here: with only a Ricci lower bound, we no longer have the uniform regularity in harmonic coordinates used in the L 2 L^2 L 2 [18 ] . The almost Euclidean balls in this decomposition are controlled by the local estimates of Liu and Székelyhidi [24 ] . The remaining difficulty is to control the Ricci integral on regular balls inside a neck. To sum these local bounds over all scales, we need an estimate that is linear in the entropy pinching.
This linear bound is the new estimate in Proposition 3.1 . Using a section of the canonical bundle and the cone function, we split the scalar-curvature integral into two terms. We control the weighted term by Ricci contractions through Bochner formulas. For the other term, we use products of two L 2 L^2 L 2 Proposition 2.17 converts this estimate into linear entropy control on regular balls, which can be summed using entropy monotonicity and the neck packing estimates.
§ 2 records the needed results and consequences from [6 , 24 ] . § 3 proves the key local linear estimate; the development of this argument was substantially assisted by AI, as explained in the Acknowledgments. § 4 follows the Jiang–Naber neck summation scheme to complete the proof.
Neck regions and entropy estimates
We recall the geometric decomposition and localized entropy estimates from [6 ] , together with the local Kähler results from [24 ] used in the proof. We also record several consequences of these results in the form needed below. Theorem 2.11 provides a splitting map and a cone function whose Hessian energies and Ricci contractions are controlled linearly by entropy pinching; Proposition 2.17 collects the estimates needed for the curvature calculation. All manifolds below are complete and smooth.
Symmetry, volume ratios, and neck regions
We first describe the cone models and centers of a codimension two neck. The Euclidean directions in the cone model determine the axis in the neck definition.
Definition 2.1. A ball B s ( x ) B_s(x) B s ( x ) ( k , ε ) (k,\varepsilon) ( k , ε )
X ′ = R k × C ( Z ) , x 0 = ( 0 , o ) , X' = \mathbb{R}^k \times C(Z), \qquad x_0 = (0,o), X ′ = R k × C ( Z ) , x 0 = ( 0 , o ) , where o o o
d G H ( B s ( x ) , B s ( x 0 ) ) < ε s . d_{\mathrm{GH}}(B_s(x), B_s(x_0)) < \varepsilon s. d GH ( B s ( x ) , B s ( x 0 )) < ε s . Given a corresponding GH approximation ι : B s ( x 0 ) ⟶ B s ( x ) \iota:B_s(x_0)\longrightarrow B_s(x) ι : B s ( x 0 ) ⟶ B s ( x )
L x , s = ι ( ( R k × { o } ) ∩ B s ( x 0 ) ) . L_{x,s}=\iota\left((\mathbb{R}^k\times\{o\})\cap B_s(x_0)\right). L x , s = ι ( ( R k × { o }) ∩ B s ( x 0 ) ) . We then say that the ball is symmetric with respect to L x , s L_{x,s} L x , s L x , s L_{x,s} L x , s
For Ric ≥ − ( n − 1 ) κ g \operatorname{Ric}\geq-(n-1)\kappa g Ric ≥ − ( n − 1 ) κ g κ ≥ 0 \kappa\geq0 κ ≥ 0
ν s κ ( x ) = Vol ( B s ( x ) ) Vol − κ ( B s ) . (2) \nu_s^\kappa(x)=\frac{\operatorname{Vol}(B_s(x))}{\operatorname{Vol}_{-\kappa}(B_s)}. \tag*{(2)} ν s κ ( x ) = Vol − κ ( B s ) Vol ( B s ( x )) . (2) where the denominator is the volume in the simply connected space form of curvature − κ -\kappa − κ κ \kappa κ
A neck is the region outside a family of center balls, where the space is close to codimension two cones at each scale. The conditions below are used in the packing and function estimates.
Definition 2.2 ([6 ] ). Let ( M n , g ) (M^n,g) ( M n , g ) R R R v v v δ 0 \delta_0 δ 0 η > 0 \eta>0 η > 0 κ ≥ 0 \kappa\geq0 κ ≥ 0 Ric ≥ − ( n − 1 ) κ g \operatorname{Ric}\geq-(n-1)\kappa g Ric ≥ − ( n − 1 ) κ g κ R 2 ≤ δ 0 2 \kappa R^2\leq\delta_0^2 κ R 2 ≤ δ 0 2 Vol ( B R ( q ) ) ≥ v R n \operatorname{Vol}(B_R(q))\geq vR^n Vol ( B R ( q )) ≥ v R n C ⊂ B 2 R ( q ) C\subset B_{2R}(q) C ⊂ B 2 R ( q ) q ∈ C q\in C q ∈ C x ↦ r x > 0 x\mapsto r_x>0 x ↦ r x > 0 C C C
N = B 2 R ( q ) ∖ B r x ‾ ( C ) , B r x ‾ ( C ) = ⋃ x ∈ C B r x ‾ ( x ) , (3) \mathcal{N}=B_{2R}(q)\setminus\overline{B_{r_x}}(C),\qquad\overline{B_{r_x}}(C)=\bigcup_{x\in C}\overline{B_{r_x}}(x), \tag*{(3)} N = B 2 R ( q ) ∖ B r x ( C ) , B r x ( C ) = x ∈ C ⋃ B r x ( x ) , (3) is an ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η )
(n1) The balls B τ n r x ( x ) B_{\tau_n r_x}(x) B τ n r x ( x ) x ∈ C x\in C x ∈ C B 2 R ( q ) B_{2R}(q) B 2 R ( q ) τ n = 10 − 10 n ω n \tau_n=10^{-10n}\omega_n τ n = 1 0 − 10 n ω n ω n = Vol ( B 1 ( 0 n ) ) \omega_n=\operatorname{Vol}(B_1(0^n)) ω n = Vol ( B 1 ( 0 n ))
(n2) For every x ∈ C x\in C x ∈ C
∣ ν δ 0 − 1 R ( x ) − ν δ 0 r x ( x ) ∣ < δ 0 2 . \left|\nu_{\delta_0^{-1}R}(x)-\nu_{\delta_0r_x}(x)\right|<\delta_0^2. ν δ 0 − 1 R ( x ) − ν δ 0 r x ( x ) < δ 0 2 . (n3) For x ∈ C x\in C x ∈ C r x ≤ s ≤ δ 0 − 1 R r_x\leq s\leq\delta_0^{-1}R r x ≤ s ≤ δ 0 − 1 R B s ( x ) B_s(x) B s ( x ) ( n − 2 , δ 0 2 ) (n-2,\delta_0^2) ( n − 2 , δ 0 2 ) L x , s L_{x,s} L x , s ( n − 1 , η ) (n-1,\eta) ( n − 1 , η )
(n4) For x ∈ C x\in C x ∈ C s ≥ r x s\geq r_x s ≥ r x B 2 s ( x ) ⊂ B 2 R ( q ) B_{2s}(x)\subset B_{2R}(q) B 2 s ( x ) ⊂ B 2 R ( q )
L x , s ⊂ B τ n s ( C ) , C ∩ B s ( x ) ⊂ B τ n s ( L x , s ) . L_{x,s}\subset B_{\tau_n s}(C),\qquad C\cap B_s(x)\subset B_{\tau_n s}(L_{x,s}). L x , s ⊂ B τ n s ( C ) , C ∩ B s ( x ) ⊂ B τ n s ( L x , s ) . (n5) Lip r x ≤ δ 0 \operatorname{Lip}r_x\leq\delta_0 Lip r x ≤ δ 0
Here B t ( D ) = ⋃ z ∈ D B t ( z ) B_t(D)=\bigcup_{z\in D}B_t(z) B t ( D ) = ⋃ z ∈ D B t ( z )
All center radii are positive in the smooth case. We weight each center by the ( n − 2 ) (n-2) ( n − 2 )
Definition 2.3. Let N = B 2 R ( q ) ∖ B r x ‾ ( C ) \mathcal{N}=B_{2R}(q)\setminus\overline{B_{r_x}}(C) N = B 2 R ( q ) ∖ B r x ( C ) ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η )
μ = μ N = ∑ x ∈ C r x n − 2 δ x , (4) \mu=\mu_{\mathcal{N}}=\sum_{x\in C}r_x^{n-2}\delta_x, \tag*{(4)} μ = μ N = x ∈ C ∑ r x n − 2 δ x , (4) where δ x \delta_x δ x x x x
Neck structure and neck decomposition
The following two results control the size of individual necks and the total size of the decomposition. The Ahlfors estimate for the center measure will be used to sum the local entropy bounds on each neck.
Theorem 2.4 (Neck Structure Theorem [6 ] ). Let N = B 2 R ( q ) ∖ B ‾ r x ( C ) \mathcal{N}=B_{2R}(q)\setminus\overline{B}_{r_x}(C) N = B 2 R ( q ) ∖ B r x ( C ) ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) Vol ( B R ( q ) ) ≥ v R n \operatorname{Vol}(B_R(q))\geq vR^n Vol ( B R ( q )) ≥ v R n δ 0 ≤ δ 0 ( n , v , η ) \delta_0\leq\delta_0(n,v,\eta) δ 0 ≤ δ 0 ( n , v , η ) μ = ∑ x ∈ C r x n − 2 δ x \mu=\sum_{x\in C}r_x^{n-2}\delta_x μ = ∑ x ∈ C r x n − 2 δ x
A 0 − 1 s n − 2 ≤ μ ( B s ( x ) ) ≤ A 0 s n − 2 , A 0 = A 0 ( n ) , (5) A_0^{-1}s^{n-2}\leq\mu(B_s(x))\leq A_0s^{n-2},\qquad A_0=A_0(n), \tag*{(5)} A 0 − 1 s n − 2 ≤ μ ( B s ( x )) ≤ A 0 s n − 2 , A 0 = A 0 ( n ) , (5) whenever x ∈ C x\in C x ∈ C s ≥ r x s\geq r_x s ≥ r x B 2 s ( x ) ⊂ B 2 R ( q ) B_{2s}(x)\subset B_{2R}(q) B 2 s ( x ) ⊂ B 2 R ( q )
The next theorem provides the covering used in the final step of the proof. The bound on the sum of the ( n − 2 ) (n-2) ( n − 2 )
Theorem 2.5 (Neck Decomposition Theorem [6 ] ). Let ( M n , g , p ) (M^n,g,p) ( M n , g , p ) Ric ≥ − ( n − 1 ) g \operatorname{Ric}\geq-(n-1)g Ric ≥ − ( n − 1 ) g Vol ( B 1 ( p ) ) ≥ v > 0 \operatorname{Vol}(B_1(p))\geq v>0 Vol ( B 1 ( p )) ≥ v > 0 η > 0 \eta>0 η > 0 δ 0 ≤ δ 0 ( n , v , η ) \delta_0\leq\delta_0(n,v,\eta) δ 0 ≤ δ 0 ( n , v , η )
B 1 ( p ) ⊂ ⋃ a ( N a ∩ B r a ( x a ) ) ∪ ⋃ b B r b ( x b ) , (6) B_1(p)\subset\bigcup_a\left(\mathcal{N}_a\cap B_{r_a}(x_a)\right)\cup\bigcup_b B_{r_b}(x_b), \tag*{(6)} B 1 ( p ) ⊂ a ⋃ ( N a ∩ B r a ( x a ) ) ∪ b ⋃ B r b ( x b ) , (6) where 0 < r a , r b ≤ 1 0<r_a,r_b\leq1 0 < r a , r b ≤ 1 N a ⊂ B 2 r a ( x a ) \mathcal{N}_a\subset B_{2r_a}(x_a) N a ⊂ B 2 r a ( x a ) ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) B 2 r b ( x b ) B_{2r_b}(x_b) B 2 r b ( x b ) ( n − 1 , 2 η ) (n-1,2\eta) ( n − 1 , 2 η )
∑ a r a n − 2 + ∑ b r b n − 2 ≤ C ( n , v , δ 0 , η ) . (7) \sum_a r_a^{n-2}+\sum_b r_b^{n-2}\leq C(n,v,\delta_0,\eta). \tag*{(7)} a ∑ r a n − 2 + b ∑ r b n − 2 ≤ C ( n , v , δ 0 , η ) . (7) We use codimension one rigidity to regard the latter balls as almost Euclidean on a fixed interior scale. The codimension two cone models are R n − 2 × C ( S α 1 ) \mathbb{R}^{n-2}\times C(S^1_\alpha) R n − 2 × C ( S α 1 ) α ≥ α ( n , v ) > 0 \alpha\geq\alpha(n,v)>0 α ≥ α ( n , v ) > 0
The localized pointed entropy
We use the localized pointed entropy to measure the error in the cone approximation at each scale. Its monotonicity and bounded range will control the sum of entropy differences in the neck estimate.
For this subsection assume
Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B s ( q ) ) ≥ v s n ( 0 < s ≤ δ 1 − 1 ) . (8) \operatorname{Ric}_g\geq-(n-1)\delta_1^2g,\qquad\operatorname{Vol}(B_s(q))\geq vs^n\quad(0<s\leq\delta_1^{-1}). \tag*{(8)} Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B s ( q )) ≥ v s n ( 0 < s ≤ δ 1 − 1 ) . (8) We use the same cutoff for all centers. The correction terms make the entropy monotone under the Ricci lower bound.
Definition 2.6 ([6 ] ). Let ( M n , g , q ) (M^n,g,q) ( M n , g , q ) v , δ 1 > 0 v,\delta_1>0 v , δ 1 > 0
Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q ) ) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) . \operatorname{Ric}_g\geq-(n-1)\delta_1^2g,\qquad\operatorname{Vol}(B_\rho(q))\geq v\rho^n\quad(0<\rho\leq\delta_1^{-1}). Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q )) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) . Write the heat kernel and its measure as
ρ t ( x , y ) = ( 4 π t ) − n / 2 e − f x , t ( y ) , ρ t ( x , d y ) = ρ t ( x , y ) d V g ( y ) . \rho_t(x,y)=(4\pi t)^{-n/2}e^{-f_{x,t}(y)},\qquad\rho_t(x,dy)=\rho_t(x,y)\,dV_g(y). ρ t ( x , y ) = ( 4 π t ) − n /2 e − f x , t ( y ) , ρ t ( x , d y ) = ρ t ( x , y ) d V g ( y ) . Choose 0 ≤ ϕ ≤ 1 0\leq\phi\leq1 0 ≤ ϕ ≤ 1
ϕ = 1 on B δ 1 − 1 ( q ) , supp ϕ ⊂ B 2 δ 1 − 1 ( q ) , ∣ Δ ϕ ∣ + ∣ ∇ ϕ ∣ 2 ≤ C ( n ) δ 1 2 . \phi=1\ \text{on }B_{\delta_1^{-1}}(q),\qquad\operatorname{supp}\phi\subset B_{2\delta_1^{-1}}(q),\qquad|\Delta\phi|+|\nabla\phi|^2\leq C(n)\delta_1^2. ϕ = 1 on B δ 1 − 1 ( q ) , supp ϕ ⊂ B 2 δ 1 − 1 ( q ) , ∣Δ ϕ ∣ + ∣∇ ϕ ∣ 2 ≤ C ( n ) δ 1 2 . For fixed x , t x,t x , t
W t = 2 t Δ f f + t ∣ ∇ f ∣ 2 + f − n , Δ f = Δ − ⟨ ∇ f , ∇ ⋅ ⟩ , f = f x , t . W_t=2t\Delta_f f+t|\nabla f|^2+f-n,\qquad\Delta_f=\Delta-\langle\nabla f,\nabla\cdot\rangle,\qquad f=f_{x,t}. W t = 2 t Δ f f + t ∣∇ f ∣ 2 + f − n , Δ f = Δ − ⟨ ∇ f , ∇ ⋅ ⟩ , f = f x , t . For x ∈ B δ 1 − 1 / 2 ( q ) x\in B_{\delta_1^{-1/2}}(q) x ∈ B δ 1 − 1/2 ( q ) 0 < t ≤ δ 1 − 2 0<t\leq\delta_1^{-2} 0 < t ≤ δ 1 − 2
W t δ 1 ( x ) = ∫ M W t ϕ ρ t ( x , d y ) − 4 ( n − 1 ) δ 1 2 ∫ 0 t s ∫ M ∣ ∇ f x , s ∣ 2 ϕ ρ s ( x , d y ) d s − C ( n , v ) δ 1 2 ∫ 0 t exp ( − 1 100 δ 1 2 s ) d s . (9) \mathcal{W}^{\delta_1}_t(x)=\int_M W_t\phi\,\rho_t(x,dy)
-4(n-1)\delta_1^2\int_0^t s\int_M|\nabla f_{x,s}|^2\phi\,\rho_s(x,dy)\,ds
-C(n,v)\delta_1^2\int_0^t\exp\left(-\frac{1}{100\delta_1^2s}\right)\,ds. \tag*{(9)} W t δ 1 ( x ) = ∫ M W t ϕ ρ t ( x , d y ) − 4 ( n − 1 ) δ 1 2 ∫ 0 t s ∫ M ∣∇ f x , s ∣ 2 ϕ ρ s ( x , d y ) d s − C ( n , v ) δ 1 2 ∫ 0 t exp ( − 100 δ 1 2 s 1 ) d s . (9) The constant C ( n , v ) C(n,v) C ( n , v )
The next theorem relates the entropy drop to squared Hessian errors and Ricci contractions. It also compares entropy with the logarithmic volume ratio when the latter changes little across scales.
Theorem 2.7 ([6 ] ). Let ( M n , g , q ) (M^n,g,q) ( M n , g , q ) v , δ 1 > 0 v,\delta_1>0 v , δ 1 > 0
Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q ) ) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) . \operatorname{Ric}_g\geq-(n-1)\delta_1^2g,\qquad\operatorname{Vol}(B_\rho(q))\geq v\rho^n\quad(0<\rho\leq\delta_1^{-1}). Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q )) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) . Use the localized entropy W δ 1 \mathcal{W}^{\delta_1} W δ 1 q q q x ∈ B δ 1 − 1 / 2 ( q ) x\in B_{\delta_1^{-1/2}}(q) x ∈ B δ 1 − 1/2 ( q ) 0 < t ≤ δ 1 − 2 0<t\leq\delta_1^{-2} 0 < t ≤ δ 1 − 2
d d t W t δ 1 ( x ) ≤ − 2 t ∫ M ( ∣ Hess f x , t − g 2 t ∣ 2 + Ric ( ∇ f x , t , ∇ f x , t ) + 2 ( n − 1 ) δ 1 2 ∣ ∇ f x , t ∣ 2 ) ϕ ρ t ( x , d y ) ≤ 0. (10) \frac{d}{dt}\mathcal{W}^{\delta_1}_t(x)\leq-2t\int_M\left(\left|\operatorname{Hess}f_{x,t}-\frac{g}{2t}\right|^2+\operatorname{Ric}(\nabla f_{x,t},\nabla f_{x,t})+2(n-1)\delta_1^2|\nabla f_{x,t}|^2\right)\phi\,\rho_t(x,dy)\leq0. \tag*{(10)} d t d W t δ 1 ( x ) ≤ − 2 t ∫ M ( Hess f x , t − 2 t g 2 + Ric ( ∇ f x , t , ∇ f x , t ) + 2 ( n − 1 ) δ 1 2 ∣∇ f x , t ∣ 2 ) ϕ ρ t ( x , d y ) ≤ 0. (10) Moreover, given ε > 0 \varepsilon>0 ε > 0 δ 1 ≤ δ 1 ( n , v , ε ) \delta_1\leq\delta_1(n,v,\varepsilon) δ 1 ≤ δ 1 ( n , v , ε ) 0 < t ≤ 10 0<t\leq10 0 < t ≤ 10
∣ V δ 1 − 1 t δ 1 2 ( x ) − V δ 1 t δ 1 2 ( x ) ∣ ≤ δ 1 , (11) \left|\mathcal{V}^{\delta_1^2}_{\delta_1^{-1}\sqrt{t}}(x)-\mathcal{V}^{\delta_1^2}_{\delta_1\sqrt{t}}(x)\right|\leq\delta_1, \tag*{(11)} V δ 1 − 1 t δ 1 2 ( x ) − V δ 1 t δ 1 2 ( x ) ≤ δ 1 , (11) then
∣ W t δ 1 ( x ) − log V t δ 1 2 ( x ) ∣ ≤ ε . (12) \left|\mathcal{W}^{\delta_1}_t(x)-\log\mathcal{V}^{\delta_1^2}_{\sqrt{t}}(x)\right|\leq\varepsilon. \tag*{(12)} W t δ 1 ( x ) − log V t δ 1 2 ( x ) ≤ ε . (12) The bounds below and monotonicity allow us to sum the entropy drops over all scales.
Remark 2.8. The heat kernel estimates and monotonicity give, for the same centers and times,
− C ( n , v ) ≤ W t δ 1 ( x ) ≤ 0 , lim t ↓ 0 W t δ 1 ( x ) = 0. (13) -C(n,v)\leq\mathcal{W}^{\delta_1}_t(x)\leq0,\qquad\lim_{t\downarrow0}\mathcal{W}^{\delta_1}_t(x)=0. \tag*{(13)} − C ( n , v ) ≤ W t δ 1 ( x ) ≤ 0 , t ↓ 0 lim W t δ 1 ( x ) = 0. (13) Under g ~ = a − 2 g \widetilde{g}=a^{-2}g g = a − 2 g ϕ \phi ϕ
W a 2 t δ 1 , g , ϕ ( x ) = W t a δ 1 , g ~ , ϕ ( x ) . (14) \mathcal{W}^{\delta_1,g,\phi}_{a^2t}(x)=\mathcal{W}^{a\delta_1,\widetilde{g},\phi}_t(x). \tag*{(14)} W a 2 t δ 1 , g , ϕ ( x ) = W t a δ 1 , g , ϕ ( x ) . (14) The splitting map and the cone function
We next recall the functions used in the curvature calculation: a harmonic splitting map and a Poisson approximation to squared distance. Entropy pinching at independent centers controls the errors in all the splitting directions.
Definition 2.9. For k ≥ 1 k \ge1 k ≥ 1 U = { x 0 , … , x k } ⊂ B 2 r ( x ) U = \{x_0,\ldots,x_k\} \subset B_{2r}(x) U = { x 0 , … , x k } ⊂ B 2 r ( x ) ( k , α ) (k,\alpha) ( k , α ) r r r
d G H ( U , U ′ ) ≥ α r d_{\mathrm{GH}}(U,U') \ge\alpha r d GH ( U , U ′ ) ≥ α r for every configuration U ′ = { x 0 ′ , … , x k ′ } ⊂ R k − 1 U' = \{x'_0,\ldots,x'_k\} \subset\mathbb{R}^{k-1} U ′ = { x 0 ′ , … , x k ′ } ⊂ R k − 1 2 r 2 ≤ δ 1 − 2 2r^2 \le\delta_1^{-2} 2 r 2 ≤ δ 1 − 2 ( k , α , δ 1 ) (k,\alpha,\delta_1) ( k , α , δ 1 )
E r k , α , δ 1 ( x ) = inf { x 0 , … , x k } ∑ a = 0 k ( W r 2 δ 1 ( x a ) − W 2 r 2 δ 1 ( x a ) ) (15) E_r^{k,\alpha,\delta_1}(x) = \inf_{\{x_0,\ldots,x_k\}} \sum_{a=0}^{k} \left(\mathcal{W}_{r^2}^{\delta_1}(x_a)-\mathcal{W}_{2r^2}^{\delta_1}(x_a)\right) \tag*{(15)} E r k , α , δ 1 ( x ) = { x 0 , … , x k } inf a = 0 ∑ k ( W r 2 δ 1 ( x a ) − W 2 r 2 δ 1 ( x a ) ) (15) where the infimum is over these independent collections in B 2 r ( x ) B_{2r}(x) B 2 r ( x ) B δ 1 − 1 / 2 ( q ) B_{\delta_1^{-1/2}}(q) B δ 1 − 1/2 ( q ) q q q
The harmonic splitting map represents the Euclidean factor of the cone model. Its defining estimates express approximate orthonormality and small Hessian energy.
Definition 2.10. A map u = ( u 1 , … , u k ) : B r ( x ) → R k u=(u^1,\ldots,u^k): B_r(x)\to\mathbb{R}^k u = ( u 1 , … , u k ) : B r ( x ) → R k ( k , ϵ ) (k,\epsilon) ( k , ϵ ) Δ u a = 0 \Delta u^a=0 Δ u a = 0 a a a
max 1 ≤ a ≤ k sup B r ( x ) ∣ ∇ u a ∣ ≤ 1 + ϵ , \max_{1\leq a\leq k}\sup_{B_r(x)}|\nabla u^a| \leq1+\epsilon, 1 ≤ a ≤ k max B r ( x ) sup ∣∇ u a ∣ ≤ 1 + ϵ , \fint_{B_r(x)} \left|\langle\nabla u^a,\nabla u^b\rangle-\delta_{ab}\right|\,dV \leq\epsilon\qquad(1\leq a,b\leq k),
r^2\fint_{B_r(x)}\sum_{a=1}^{k}|\operatorname{Hess}u^a|^2\,dV \leq\epsilon^2.
We next pass from pinching averaged over the neck centers to pinching at an independent collection of centers. The same center-selection argument appears in Cheeger–Jiang–Naber and Jiang–Naber. We record the resulting consequence and omit the proof.
The next theorem provides a splitting map and a cone function with linear bounds for their Hessian energies and Ricci contractions. After rescaling, the following result follows from the sharp cone-splitting and Poisson regularization estimates.
Theorem 2.11 [6 ] . Given n ≥ 3 n \ge3 n ≥ 3 v , η , ε , α > 0 v,\eta,\varepsilon,\alpha> 0 v , η , ε , α > 0 δ 0 ≤ δ 0 ( n , v , α , ε ) \delta_0 \le\delta_0(n,v,\alpha,\varepsilon) δ 0 ≤ δ 0 ( n , v , α , ε )
N = B 2 R ( q ) ∖ B ‾ r x ( C ) \mathcal{N}=B_{2R}(q)\setminus\overline{B}_{r_x}(\mathcal{C}) N = B 2 R ( q ) ∖ B r x ( C ) be a smooth ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) r q ≤ R r_q \le R r q ≤ R ( M n , g ) (M^n,g) ( M n , g ) δ 1 = 4 δ 0 / R \delta_1=4\delta_0/R δ 1 = 4 δ 0 / R
Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q ) ) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) . \operatorname{Ric}_g \ge-(n-1)\delta_1^2g,\qquad\operatorname{Vol}(B_\rho(q)) \ge v\rho^n\quad(0<\rho\le\delta_1^{-1}). Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q )) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) . Use the localized entropy with one fixed cutoff based at q q q ( n − 2 , ε ) (n-2,\varepsilon) ( n − 2 , ε ) u = ( u 1 , … , u n − 2 ) : B 2 R ( q ) → R n − 2 u=(u^1,\ldots,u^{n-2}):B_{2R}(q)\to\mathbb{R}^{n-2} u = ( u 1 , … , u n − 2 ) : B 2 R ( q ) → R n − 2 h : B 2 R ( q ) → R h:B_{2R}(q)\to\mathbb{R} h : B 2 R ( q ) → R
R^2\mathop{\fint}_{B_{2R}(q)}\sum_{a=1}^{n-2}\left(\lvert\operatorname{Hess}u^a\rvert^2+\operatorname{Ric}(\nabla u^a,\nabla u^a)+2(n-1)\delta_1^2\lvert\nabla u^a\rvert^2\right)\,dV
\le C(n,v,\alpha)E_R^{n-2,\alpha,\delta_1}(q). \tag*{(18)}
Δ h = 2 n on B 2 R ( q ) , (19) \Delta h=2n\qquad\text{on }B_{2R}(q), \tag*{(19)} Δ h = 2 n on B 2 R ( q ) , (19) \mathop{\fint}_{B_{2R}(q)}\left(\lvert\operatorname{Hess}h-2g\rvert^2+\operatorname{Ric}(\nabla h,\nabla h)+2(n-1)\delta_1^2\lvert\nabla h\rvert^2\right)\,dV
\le C(n,v)\left(\mathcal{W}_{R^2}^{\delta_1}(q)-\mathcal{W}_{2R^2}^{\delta_1}(q)\right), \tag*{(20)}
\mathop{\fint}_{B_{2R}(q)}\left\lvert\nabla h\right\rvert^2-4h\,dV
\le C(n,v)R^4\left(\mathcal{W}_{R^2}^{\delta_1}(q)-\mathcal{W}_{2R^2}^{\delta_1}(q)\right), \tag*{(21)}
and
sup B 2 R ( q ) ∣ ∇ h ∣ ≤ C ( n , v ) R , sup B 2 R ( q ) ∣ h − d q 2 ∣ ≤ ε R 2 . (22) \sup_{B_{2R}(q)}\lvert\nabla h\rvert\le C(n,v)R,\qquad
\sup_{B_{2R}(q)}\lvert h-d_q^2\rvert\le\varepsilon R^2. \tag*{(22)} B 2 R ( q ) sup ∣ ∇ h ∣ ≤ C ( n , v ) R , B 2 R ( q ) sup ∣ h − d q 2 ∣ ≤ ε R 2 . (22) The neck hypotheses provide the noncollapsing needed for this localization, after decreasing v v v u , h u,h u , h P P P h − ∣ u ∣ 2 h-\lvert u\rvert^2 h − ∣ u ∣ 2 y y y
Corollary 2.12. Let N ⊂ B 2 R ( q ) \mathcal{N}\subset B_{2R}(q) N ⊂ B 2 R ( q ) ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) n ≥ 3 n\ge3 n ≥ 3 v , δ 1 > 0 v,\delta_1>0 v , δ 1 > 0
Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q ) ) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) , δ 1 R ≤ 4 δ 0 ≤ 1 / 100. \operatorname{Ric}_g\ge-(n-1)\delta_1^2g,\qquad\operatorname{Vol}(B_\rho(q))\ge v\rho^n\quad(0<\rho\le\delta_1^{-1}),\qquad\delta_1R\le4\delta_0\le1/100. Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q )) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) , δ 1 R ≤ 4 δ 0 ≤ 1/100. Fix L ≥ L 0 ( n , v , η ) ≥ 32 L\ge L_0(n,v,\eta)\ge32 L ≥ L 0 ( n , v , η ) ≥ 32
y ∈ N ∩ B R ( q ) , z ∈ C , r = d ( y , C ) = d ( y , z ) > 0 , s = L r ≤ 10 − 6 R . y\in\mathcal{N}\cap B_R(q),\qquad z\in\mathcal{C},\qquad r=d(y,\mathcal{C})=d(y,z)>0,\qquad s=Lr\le10^{-6}R. y ∈ N ∩ B R ( q ) , z ∈ C , r = d ( y , C ) = d ( y , z ) > 0 , s = L r ≤ 1 0 − 6 R . Let μ = μ N \mu=\mu_{\mathcal{N}} μ = μ N P P P (16) , using one fixed entropy cutoff based at q q q
k = n − 2 , κ = ( n − 1 ) δ 1 2 , R = Ric + κ g ≥ 0. k=n-2,\qquad\kappa=(n-1)\delta_1^2,\qquad\mathcal{R}=\operatorname{Ric}+\kappa g\ge0. k = n − 2 , κ = ( n − 1 ) δ 1 2 , R = Ric + κ g ≥ 0. There are ε 0 > 0 \varepsilon_0 > 0 ε 0 > 0 C < ∞ C < \infty C < ∞ 0 < c ≤ C < ∞ 0 < c \le C < \infty 0 < c ≤ C < ∞ n , v , η , L n,v,\eta,L n , v , η , L P + δ 0 ≤ ε 0 P+\delta_0 \le\varepsilon_0 P + δ 0 ≤ ε 0 u 1 , … , u k u^1,\ldots,u^k u 1 , … , u k h h h B 8 s ( z ) B_{8s}(z) B 8 s ( z )
Δ h = 2 n , ∣ u ∣ ≤ C s , ∣ d u ∣ ≤ C , ∣ h ∣ ≤ C s 2 , ∣ d h ∣ ≤ C s , \Delta h=2n,\qquad|u|\le Cs,\qquad|du|\le C,\qquad|h|\le Cs^2,\qquad|dh|\le Cs, Δ h = 2 n , ∣ u ∣ ≤ C s , ∣ d u ∣ ≤ C , ∣ h ∣ ≤ C s 2 , ∣ d h ∣ ≤ C s , such that
s − n ∫ B 8 s ( z ) [ ∑ a , b = 1 k ∣ ⟨ d u a , d u b ⟩ − δ a b ∣ 2 + s 2 ∑ a = 1 k ( ∣ Hess u a ∣ 2 + R ( ∇ u a , ∇ u a ) ) + ∣ Hess h − 2 g ∣ 2 + R ( ∇ h , ∇ h ) + s − 4 ( ∣ d h ∣ 2 − 4 h ) 2 ] d V g ≤ C P . (23) s^{-n}\int_{B_{8s}(z)}\left[\sum_{a,b=1}^{k}\left|\langle du^a,du^b\rangle-\delta_{ab}\right|^2+s^2\sum_{a=1}^{k}\left(|\operatorname{Hess}u^a|^2+\mathcal{R}(\nabla u^a,\nabla u^a)\right)+|\operatorname{Hess}h-2g|^2+\mathcal{R}(\nabla h,\nabla h)+s^{-4}(|dh|^2-4h)^2\right]\,dV_g\le CP. \tag*{(23)} s − n ∫ B 8 s ( z ) a , b = 1 ∑ k ⟨ d u a , d u b ⟩ − δ ab 2 + s 2 a = 1 ∑ k ( ∣ Hess u a ∣ 2 + R ( ∇ u a , ∇ u a ) ) + ∣ Hess h − 2 g ∣ 2 + R ( ∇ h , ∇ h ) + s − 4 ( ∣ d h ∣ 2 − 4 h ) 2 d V g ≤ C P . (23) After adding constants to u a u^a u a
ℓ a = ⟨ d h , d u a ⟩ − 2 u a , s − n − 2 ∫ B 2 s ( z ) ∑ a = 1 k ℓ a 2 d V g ≤ C P . (24) \ell_a=\langle dh,du^a\rangle-2u^a,\qquad s^{-n-2}\int_{B_{2s}(z)}\sum_{a=1}^{k}\ell_a^2\,dV_g\le CP. \tag*{(24)} ℓ a = ⟨ d h , d u a ⟩ − 2 u a , s − n − 2 ∫ B 2 s ( z ) a = 1 ∑ k ℓ a 2 d V g ≤ C P . (24) For b = h − ∣ u ∣ 2 b=h-|u|^2 b = h − ∣ u ∣ 2
c s 2 ≤ b ≤ C s 2 on B r / 8 ( y ) . (25) cs^2\le b\le Cs^2\qquad\text{on }B_{r/8}(y). \tag*{(25)} c s 2 ≤ b ≤ C s 2 on B r /8 ( y ) . (25) Proof. Use the fixed-cutoff local construction underlying Theorem 2.11 . Apply it to the selected centers on a common fixed enlargement of B 8 s ( z ) B_{8s}(z) B 8 s ( z ) h h h x 0 x_0 x 0 [ s 2 / 2 , 2 s 2 ] [s^2/2,2s^2] [ s 2 /2 , 2 s 2 ] (23) ; the enlargement changes only the constants.
To prove (24) , differentiate:
d ℓ a = ( Hess h − 2 g ) ( ∇ u a , ⋅ ) + Hess u a ( ∇ h , ⋅ ) . d\ell_a=(\operatorname{Hess}h-2g)(\nabla u^a,\cdot)+\operatorname{Hess}u^a(\nabla h,\cdot). d ℓ a = ( Hess h − 2 g ) ( ∇ u a , ⋅ ) + Hess u a ( ∇ h , ⋅ ) . Replace u a u^a u a u a + 1 2 f B 2 s ( z ) ℓ a d V g u^a+\frac{1}{2}f_{B_{2s}(z)}\ell_a\,dV_g u a + 2 1 f B 2 s ( z ) ℓ a d V g ℓ a \ell_a ℓ a
∫ B 2 s ( z ) ∑ a ℓ a 2 d V g ≤ C s 2 ∫ B 4 s ( z ) ∑ a ∣ d ℓ a ∣ 2 d V g ≤ C s 2 ∫ B 4 s ( z ) ( ∣ Hess h − 2 g ∣ 2 + s 2 ∑ a ∣ Hess u a ∣ 2 ) d V g ≤ C s n + 2 P . \begin{aligned}
\int_{B_{2s}(z)}\sum_a\ell_a^2\,dV_g\le Cs^2\int_{B_{4s}(z)}\sum_a|d\ell_a|^2\,dV_g \\
&\le Cs^2\int_{B_{4s}(z)}\left(|\operatorname{Hess}h-2g|^2+s^2\sum_a|\operatorname{Hess}u^a|^2\right)\,dV_g\le Cs^{n+2}P.
\end{aligned} ∫ B 2 s ( z ) a ∑ ℓ a 2 d V g ≤ C s 2 ∫ B 4 s ( z ) a ∑ ∣ d ℓ a ∣ 2 d V g ≤ C s 2 ∫ B 4 s ( z ) ( ∣ Hess h − 2 g ∣ 2 + s 2 a ∑ ∣ Hess u a ∣ 2 ) d V g ≤ C s n + 2 P . In the normalized cone limit, h = ∣ Z − Z 0 ∣ 2 + ρ 2 h=|Z-Z_0|^2+\rho^2 h = ∣ Z − Z 0 ∣ 2 + ρ 2 u = O Z + c u=OZ+c u = O Z + c c = − O Z 0 c=-OZ_0 c = − O Z 0 b → ρ 2 b\to\rho^2 b → ρ 2 ρ ≥ r / 4 \rho\ge r/4 ρ ≥ r /4 B r / 8 ( y ) B_{r/8}(y) B r /8 ( y ) (25) after L L L
Holomorphic charts and Ricci estimates on almost Euclidean balls
The preceding results apply to general Riemannian manifolds. We now state the holomorphic chart and Ricci integral estimates used below. In this subsection, let ( M n , g , J ) (M^n,g,J) ( M n , g , J ) n = 2 m ≥ 4 n=2m\ge4 n = 2 m ≥ 4 J J J ω ( X , Y ) = g ( J X , Y ) \omega(X,Y)=g(JX,Y) ω ( X , Y ) = g ( J X , Y ) ∇ J = 0 \nabla J=0 ∇ J = 0 S = tr R Ric g S=\operatorname{tr}_{\mathbb{R}}\operatorname{Ric}_g S = tr R Ric g z = ( z 1 , … , z m ) z=(z_1,\ldots,z_m) z = ( z 1 , … , z m )
ω = i 2 ∑ a , b g a b ˉ d z a ∧ d z ˉ b , Ric ( ω ) = − i 2 π ∂ ∂ ˉ log det ( g a b ˉ ) . \omega=\frac{i}{2}\sum_{a,b}g_{a\bar b}\,dz_a\wedge d\bar z_b,\qquad\operatorname{Ric}(\omega)=-\frac{i}{2\pi}\partial\bar\partial\log\det(g_{a\bar b}). ω = 2 i a , b ∑ g a b ˉ d z a ∧ d z ˉ b , Ric ( ω ) = − 2 π i ∂ ∂ ˉ log det ( g a b ˉ ) . With this normalization,
2 π Ric ( ω ) ( X , Y ) = Ric g ( J X , Y ) . 2\pi\operatorname{Ric}(\omega)(X,Y)=\operatorname{Ric}_{g}(JX,Y). 2 π Ric ( ω ) ( X , Y ) = Ric g ( J X , Y ) . The symbol Ric g \operatorname{Ric}_{g} Ric g Ric \operatorname{Ric} Ric ω \omega ω ∇ \nabla ∇ D = D ′ + D ′ ′ D=D^{\prime}+D^{\prime\prime} D = D ′ + D ′′ D ′ ′ = ∂ ˉ D^{\prime\prime}=\bar{\partial} D ′′ = ∂ ˉ D ′ ∂ f = ( Hess f ) 2 , 0 D^{\prime}\partial f=(\operatorname{Hess}f)^{2,0} D ′ ∂ f = ( Hess f ) 2 , 0 f f f
We also write
ω 0 = i 2 ∑ j d z j ∧ d z ˉ j , d V 0 = ω 0 m m ! . \omega_{0}=\frac{i}{2}\sum_{j} dz_{j}\wedge d\bar{z}_{j},\qquad dV_{0}=\frac{\omega_{0}^{m}}{m!}. ω 0 = 2 i j ∑ d z j ∧ d z ˉ j , d V 0 = m ! ω 0 m . Here B ρ ( y ) B_{\rho}(y) B ρ ( y ) B ρ 0 ( 0 ) B_{\rho}^{0}(0) B ρ 0 ( 0 ) g g g § 3 .
The next theorem gives holomorphic coordinates and a bounded Kähler potential on almost Euclidean balls. We use these in § 3 .
Theorem 2.13 ([24 ] ). For n = 2 m ≥ 4 n=2m\geq4 n = 2 m ≥ 4 ε 0 ( n ) > 0 \varepsilon_{0}(n)>0 ε 0 ( n ) > 0 0 < ε ≤ ε 0 ( n ) 0<\varepsilon\leq\varepsilon_{0}(n) 0 < ε ≤ ε 0 ( n ) r > 0 r>0 r > 0 B ε − 1 r ( x ) B_{\varepsilon^{-1}r}(x) B ε − 1 r ( x ) n = 2 m ≥ 4 n=2m\geq4 n = 2 m ≥ 4
Ric g > − ε r − 2 g , d G H ( B ε − 1 r ( x ) , B ε − 1 r ( 0 n ) ) < ε r . \operatorname{Ric}_{g}>-\varepsilon r^{-2}g,\qquad d_{\mathrm{GH}}\left(B_{\varepsilon^{-1}r}(x),B_{\varepsilon^{-1}r}(0^{n})\right)<\varepsilon r. Ric g > − ε r − 2 g , d GH ( B ε − 1 r ( x ) , B ε − 1 r ( 0 n ) ) < ε r . Then there is a holomorphic coordinate map F : B r ( x ) ⟶ C m F:B_{r}(x)\longrightarrow\mathbb{C}^{m} F : B r ( x ) ⟶ C m Ψ ( ε ∣ n ) r \Psi(\varepsilon\mid n)r Ψ ( ε ∣ n ) r Ψ ( ε ∣ n ) → 0 \Psi(\varepsilon\mid n)\to0 Ψ ( ε ∣ n ) → 0 ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 ρ \rho ρ B r ( x ) B_{r}(x) B r ( x )
ω = i ∂ ∂ ˉ ρ , sup B r ( x ) ∣ ρ ∣ ≤ C ( n ) r 2 . \omega=i\partial\bar{\partial}\rho,\qquad\sup_{B_{r}(x)}|\rho|\leq C(n)r^{2}. ω = i ∂ ∂ ˉ ρ , B r ( x ) sup ∣ ρ ∣ ≤ C ( n ) r 2 . The last estimate is the bounded-potential consequence of the potential approximation in the cited theorem.
The scalar-curvature estimate of Liu and Székelyhidi also gives the following Ricci L 1 L^{1} L 1 ∣ Ric ∣ ≤ S + 2 n κ |\operatorname{Ric}|\leq S+2n\kappa ∣ Ric ∣ ≤ S + 2 nκ Ric ≥ − κ g \operatorname{Ric}\geq-\kappa g Ric ≥ − κ g
Theorem 2.14 ([24 ] ). Given n = 2 m ≥ 4 n=2m\geq4 n = 2 m ≥ 4 v > 0 v>0 v > 0 ε = ε ( n , v ) > 0 \varepsilon=\varepsilon(n,v)>0 ε = ε ( n , v ) > 0 C = C ( n , v ) < ∞ C=C(n,v)<\infty C = C ( n , v ) < ∞ ( M n , g , J ) (M^{n},g,J) ( M n , g , J ) n n n r > 0 r>0 r > 0
Ric g ≥ − ( n − 1 ) ε r − 2 g , Vol g ( B r ( x ) ) ≥ v r n , \operatorname{Ric}_{g}\geq-(n-1)\varepsilon r^{-2}g,\qquad\operatorname{Vol}_{g}(B_{r}(x))\geq vr^{n}, Ric g ≥ − ( n − 1 ) ε r − 2 g , Vol g ( B r ( x )) ≥ v r n , d G H ( B 4 r ( x ) , B 4 r ( 0 n ) ) < ε r , d_{\mathrm{GH}}\left(B_{4r}(x),B_{4r}(0^{n})\right)<\varepsilon r, d GH ( B 4 r ( x ) , B 4 r ( 0 n ) ) < ε r , then
∫ B r ( x ) ∣ Ric g ∣ d V g ≤ C r n − 2 . \int_{B_{r}(x)}|\operatorname{Ric}_{g}|\,dV_{g}\leq Cr^{n-2}. ∫ B r ( x ) ∣ Ric g ∣ d V g ≤ C r n − 2 . Since the neck models are flat away from their axes, Theorem 2.13 and the interior gradient estimate for its harmonic coordinate functions apply on regular balls. The following corollary gives the chart, potential, and metric bounds needed for the holomorphic correction and curvature integral in § 3 .
Corollary 2.15. For n = 2 m ≥ 4 n=2m\ge4 n = 2 m ≥ 4 v , η > 0 v,\eta>0 v , η > 0 0 < ε < 1 / 100 0<\varepsilon<1/100 0 < ε < 1/100 a 0 ∈ ( 0 , 1 / 8 ) a_0\in(0,1/8) a 0 ∈ ( 0 , 1/8 ) δ > 0 \delta>0 δ > 0 C < ∞ C<\infty C < ∞ n , v , η , ε n,v,\eta,\varepsilon n , v , η , ε N ⊂ B 2 R ( q ) \mathcal{N}\subset B_{2R}(q) N ⊂ B 2 R ( q ) ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) n = 2 m ≥ 4 n=2m\ge4 n = 2 m ≥ 4 δ 0 ≤ δ \delta_0\le\delta δ 0 ≤ δ Vol g ( B R ( q ) ) ≥ v R n \operatorname{Vol}_g(B_R(q))\ge vR^n Vol g ( B R ( q )) ≥ v R n y ∈ N ∩ B R ( q ) y\in\mathcal{N}\cap B_R(q) y ∈ N ∩ B R ( q ) r = d ( y , C ) > 0 r=d(y,\mathcal{C})>0 r = d ( y , C ) > 0
z = ( z 1 , … , z m ) : B a 0 r ( y ) ⟶ C m , z ( y ) = 0 , z=(z_1,\ldots,z_m):B_{a_0r}(y)\longrightarrow\mathbb{C}^m,\qquad z(y)=0, z = ( z 1 , … , z m ) : B a 0 r ( y ) ⟶ C m , z ( y ) = 0 , which, for the rescaled metric g ~ = r − 2 g \widetilde{g}=r^{-2}g g = r − 2 g ε a 0 \varepsilon a_0 ε a 0 g ~ \widetilde{g} g ρ ~ \widetilde{\rho} ρ
∣ ρ ~ ∣ ≤ C , g ~ ≥ C − 1 z ∗ g C m . |\widetilde{\rho}|\le C,\qquad\widetilde{g}\ge C^{-1}z^*g_{\mathbb{C}^m}. ∣ ρ ∣ ≤ C , g ≥ C − 1 z ∗ g C m . For the application in § 4 , choose the chart accuracy once in terms of n , v , η n,v,\eta n , v , η Theorem 2.14 and the interior ball inclusions below. Decrease the associated radius constant by a factor of 8 8 8 a 0 a_0 a 0 a 0 = a 0 ( n , v , η ) < 1 / 64 a_0=a_0(n,v,\eta)<1/64 a 0 = a 0 ( n , v , η ) < 1/64 B 8 a 0 r ( y ) B_{8a_0r}(y) B 8 a 0 r ( y ) r − 2 g r^{-2}g r − 2 g 10 − 3 a 0 10^{-3}a_0 1 0 − 3 a 0 n , v , η n,v,\eta n , v , η
Complex pairing of the splitting functions
We combine pairs of real splitting functions into nearly holomorphic functions and collect their estimates at the scale of a regular ball. Proposition 2.17 provides the energy bounds used to apply the local estimate in § 4 .
The next lemma gives a linear bound for the squared antiholomorphic derivatives after a constant orthogonal change of the real splitting functions. We state it at unit scale; the following proposition applies it at scale s = L r s=Lr s = L r
Lemma 2.16. Fix n = 2 m ≥ 4 n=2m\ge4 n = 2 m ≥ 4 v , η > 0 v,\eta>0 v , η > 0 C 0 ≥ 1 C_0\ge1 C 0 ≥ 1 k = n − 2 k=n-2 k = n − 2
N = B 2 R ( q ) ∖ B r x ( C ) ‾ \mathcal{N}=B_{2R}(q)\setminus\overline{B_{r_x}(\mathcal{C})} N = B 2 R ( q ) ∖ B r x ( C ) be an ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) ( M n , g , J ) (M^n,g,J) ( M n , g , J ) n n n z ∈ C z\in\mathcal{C} z ∈ C
r z ≤ 1 / 32 , B 16 ( z ) ⊂ B 2 R ( q ) , Ric g ≥ − ε g , 0 ≤ ε ≤ 1 , Vol ( B 1 ( z ) ) ≥ v . r_z\le1/32,\qquad B_{16}(z)\subset B_{2R}(q),\qquad\operatorname{Ric}_g\ge-\varepsilon g,\qquad0\le\varepsilon\le1,\qquad\operatorname{Vol}(B_1(z))\ge v. r z ≤ 1/32 , B 16 ( z ) ⊂ B 2 R ( q ) , Ric g ≥ − ε g , 0 ≤ ε ≤ 1 , Vol ( B 1 ( z )) ≥ v . Let u 1 , … , u k , h u^1,\ldots,u^k,h u 1 , … , u k , h B 8 ( z ) B_8(z) B 8 ( z )
Δ u a = 0 , Δ h = 2 n , ∣ u ∣ + ∣ d u ∣ + ∣ h ∣ + ∣ d h ∣ ≤ C 0 . \Delta u^a=0,\qquad\Delta h=2n,\qquad|u|+|du|+|h|+|dh|\le C_0. Δ u a = 0 , Δ h = 2 n , ∣ u ∣ + ∣ d u ∣ + ∣ h ∣ + ∣ d h ∣ ≤ C 0 . Set
R = Ric + ε g , b = h − ∣ u ∣ 2 , ℓ a = ⟨ d h , d u a ⟩ − 2 u a , \mathcal{R}=\operatorname{Ric}+\varepsilon g,\qquad b=h-|u|^2,\qquad\ell_a=\langle dh,du^a\rangle-2u^a, R = Ric + ε g , b = h − ∣ u ∣ 2 , ℓ a = ⟨ d h , d u a ⟩ − 2 u a , and define F F F (23) and (24) , evaluated with s = 1 s=1 s = 1 R \mathcal{R} R ε 0 = ε 0 ( n , v , η , C 0 ) > 0 \varepsilon_0=\varepsilon_0(n,v,\eta,C_0)>0 ε 0 = ε 0 ( n , v , η , C 0 ) > 0 C = C ( n , v , η , C 0 ) < ∞ C=C(n,v,\eta,C_0)<\infty C = C ( n , v , η , C 0 ) < ∞ δ 0 + F + ε ≤ ε 0 \delta_0+F+\varepsilon\le\varepsilon_0 δ 0 + F + ε ≤ ε 0 O ∈ O ( k ) O\in O(k) O ∈ O ( k ) u u u O u Ou O u
w j ′ = u 2 j − 1 + i u 2 j , 1 ≤ j ≤ m − 1 , w'_j=u^{2j-1}+iu^{2j},\qquad1\le j\le m-1, w j ′ = u 2 j − 1 + i u 2 j , 1 ≤ j ≤ m − 1 , satisfy
∑ j = 1 m − 1 ∫ B 1 ( z ) ∣ ∂ ‾ w j ′ ∣ 2 d V g ≤ C ( F + ε ) . (26) \sum_{j=1}^{m-1}\int_{B_1(z)}|\overline{\partial}w'_j|^2\,dV_g\le C(F+\varepsilon). \tag*{(26)} j = 1 ∑ m − 1 ∫ B 1 ( z ) ∣ ∂ w j ′ ∣ 2 d V g ≤ C ( F + ε ) . (26) This change preserves b b b ∑ a ℓ a 2 \sum_a\ell_a^2 ∑ a ℓ a 2
Proof. After choosing ε 0 \varepsilon_0 ε 0 γ = γ ( n , v , η ) > 0 \gamma=\gamma(n,v,\eta)>0 γ = γ ( n , v , η ) > 0 B 1 / 2 ( z ) B_{1/2}(z) B 1/2 ( z ) ( k + 1 , γ ) (k+1,\gamma) ( k + 1 , γ ) r z ≤ 1 / 32 r_z\le1/32 r z ≤ 1/32 B 16 ( z ) ⊂ B 2 R ( q ) B_{16}(z)\subset B_{2R}(q) B 16 ( z ) ⊂ B 2 R ( q )
Write X a = ∇ u a X_a=\nabla u^a X a = ∇ u a v ^ a = 1 2 ⟨ J X a , ∇ h ⟩ \widehat v_a=\frac{1}{2}\langle JX_a,\nabla h\rangle v a = 2 1 ⟨ J X a , ∇ h ⟩
d v ^ a − ( J X a ) b = 1 2 ( Hess h − 2 g ) ( J X a , ⋅ ) + 1 2 ⟨ J Hess u a , ∇ h ⟩ , ∫ B 7 ( z ) ∣ d v ^ a − ( J X a ) b ∣ 2 d V g ≤ C F . d\widehat v_a-(JX_a)^b=\frac{1}{2}(\operatorname{Hess}h-2g)(JX_a,\cdot)+\frac{1}{2}\langle J\operatorname{Hess}u^a,\nabla h\rangle,
\qquad
\int_{B_7(z)}\left|d\widehat v_a-(JX_a)^b\right|^2dV_g\le CF. d v a − ( J X a ) b = 2 1 ( Hess h − 2 g ) ( J X a , ⋅ ) + 2 1 ⟨ J Hess u a , ∇ h ⟩ , ∫ B 7 ( z ) d v a − ( J X a ) b 2 d V g ≤ C F . Let v a v_a v a B 6 ( z ) B_6(z) B 6 ( z ) B 7 ( z ) B_7(z) B 7 ( z ) ξ a = d v a − ( J X a ) b \xi_a=dv_a-(JX_a)^b ξ a = d v a − ( J X a ) b div ( J X a ) = 0 \operatorname{div}(JX_a)=0 div ( J X a ) = 0 ∫ ∣ ξ a ∣ 2 ≤ C F \int|\xi_a|^2\le CF ∫ ∣ ξ a ∣ 2 ≤ C F J J J ξ a \xi_a ξ a
∫ B 3 ( z ) ( ∣ ξ a ∣ 2 + ∣ ∇ ξ a ∣ 2 ) d V g + sup B 2 ( z ) ∣ ξ a ∣ 2 ≤ C F . \int_{B_3(z)}\left(|\xi_a|^2+|\nabla\xi_a|^2\right)dV_g+\sup_{B_2(z)}|\xi_a|^2\le CF. ∫ B 3 ( z ) ( ∣ ξ a ∣ 2 + ∣∇ ξ a ∣ 2 ) d V g + B 2 ( z ) sup ∣ ξ a ∣ 2 ≤ C F . Define the constant skew matrix and residual forms by
\overline K_{ab}=\fint_{B_2(z)}\langle JX_a,X_b\rangle dV_g,
\qquad
\beta_a=(JX_a)^b-\sum_b\overline K_{ab}\,du^b,
\qquad
A_a^2=\fint_{B_2(z)}|\beta_a|^2dV_g.
These forms are Hodge harmonic and A a ≤ C A_a\le C A a ≤ C
∫ B 4 ( z ) ∣ ∇ β a ∣ 2 d V g + ∑ b ∫ B 2 ( z ) ∣ ⟨ β a , d u b ⟩ ∣ 2 d V g + ∫ B 2 ( z ) ( ∣ β a ∣ − A a ) 2 d V g ≤ C F , \int_{B_4(z)}|\nabla\beta_a|^2dV_g+\sum_b\int_{B_2(z)}|\langle\beta_a,du^b\rangle|^2dV_g+\int_{B_2(z)}(|\beta_a|-A_a)^2dV_g\le CF, ∫ B 4 ( z ) ∣∇ β a ∣ 2 d V g + b ∑ ∫ B 2 ( z ) ∣ ⟨ β a , d u b ⟩ ∣ 2 d V g + ∫ B 2 ( z ) ( ∣ β a ∣ − A a ) 2 d V g ≤ C F , sup B 1 ( z ) ∣ β a ∣ ≤ C A a . \sup_{B_1(z)}|\beta_a|\le CA_a. B 1 ( z ) sup ∣ β a ∣ ≤ C A a . For the cross terms use ⟨ β a , d u b ⟩ = ⟨ J X a , X b ⟩ − K ‾ a b − ∑ c K ‾ a c ( ⟨ X c , X b ⟩ − δ c b ) \langle\beta_a,du^b\rangle=\langle JX_a,X_b\rangle-\overline K_{ab}-\sum_c\overline K_{ac}(\langle X_c,X_b\rangle-\delta_{cb}) ⟨ β a , d u b ⟩ = ⟨ J X a , X b ⟩ − K ab − ∑ c K a c (⟨ X c , X b ⟩ − δ c b )
If A a > 0 A_a>0 A a > 0 τ = ( F + ε ) / A a 2 \tau=(F+\varepsilon)/A_a^2 τ = ( F + ε ) / A a 2 τ \tau τ
w = v a − ∑ b K ‾ a b u b A a , d w = β a + ξ a A a w=\frac{v_a-\sum_b\overline K_{ab}u^b}{A_a},
\qquad
dw=\frac{\beta_a+\xi_a}{A_a} w = A a v a − ∑ b K ab u b , d w = A a β a + ξ a gives a harmonic map U = ( u , w ) U=(u,w) U = ( u , w )
∫ B 1 ( z ) [ ∑ i , j ∣ ⟨ d U i , d U j ⟩ − δ i j ∣ 2 + ∑ i ∣ Hess U i ∣ 2 ] d V g ≤ C τ , sup B 1 ( z ) ∣ d U i ∣ ≤ C . \int_{B_1(z)}\left[\sum_{i,j}|\langle dU^i,dU^j\rangle-\delta_{ij}|^2+\sum_i|\operatorname{Hess}U^i|^2\right]dV_g\le C\tau,
\qquad
\sup_{B_1(z)}|dU^i|\le C. ∫ B 1 ( z ) [ i , j ∑ ∣ ⟨ d U i , d U j ⟩ − δ ij ∣ 2 + i ∑ ∣ Hess U i ∣ 2 ] d V g ≤ C τ , B 1 ( z ) sup ∣ d U i ∣ ≤ C . Here the normalized residual estimates follow by dividing by A a A_a A a C F ≤ C τ CF\le C\tau C F ≤ C τ Δ ∣ d U i ∣ 2 ≥ − C ε \Delta|dU^i|^2\ge-C\varepsilon Δ∣ d U i ∣ 2 ≥ − C ε sup B 1 / 2 ( z ) ∣ d U i ∣ ≤ 1 + C τ \sup_{B_{1/2}(z)}|dU^i|\le1+C\sqrt{\tau} sup B 1/2 ( z ) ∣ d U i ∣ ≤ 1 + C τ ε ≤ A a 2 τ ≤ C τ \varepsilon\le A_a^2\tau\le C\tau ε ≤ A a 2 τ ≤ C τ U U U ( k + 1 , C τ ) (k+1,C\sqrt{\tau}) ( k + 1 , C τ ) ∑ a A a 2 ≤ C ( F + ε ) \sum_a A_a^2\le C(F+\varepsilon) ∑ a A a 2 ≤ C ( F + ε )
Comparing the Gram matrices of J X a JX_a J X a ∑ b K ‾ a b X b \sum_b\overline K_{ab}X_b ∑ b K ab X b
∣ K ‾ K ‾ t − Id k ∣ 2 ≤ C ( F + ε ) . \left|\overline K\overline K^t-\operatorname{Id}_k\right|^2\le C(F+\varepsilon). K K t − Id k 2 ≤ C ( F + ε ) . Its polar part K 0 = K ‾ ( − K ‾ 2 ) − 1 / 2 K_0=\overline{K}(-\overline{K}^{2})^{-1/2} K 0 = K ( − K 2 ) − 1/2 ∣ K 0 − K ‾ ∣ 2 ≤ C ( F + ε ) |K_0-\overline{K}|^2\le C(F+\varepsilon) ∣ K 0 − K ∣ 2 ≤ C ( F + ε ) K 0 K_0 K 0
∑ j ∫ B 1 ( z ) ∣ ∇ u 2 j − 1 − ∇ u 2 j ∣ 2 d V g ≤ C ( F + ε ) , \sum_j\int_{B_1(z)}|\nabla u^{2j-1}-\nabla u^{2j}|^2\,dV_g\le C(F+\varepsilon), j ∑ ∫ B 1 ( z ) ∣∇ u 2 j − 1 − ∇ u 2 j ∣ 2 d V g ≤ C ( F + ε ) , which is (26) . Orthogonality preserves ∣ u ∣ 2 |u|^2 ∣ u ∣ 2
We return to the original metric and functions, keeping the notation w j ′ w'_j w j ′ Lemma 2.16 . The following proposition combines the pairing estimate with Corollary 2.12; its energy bound controls the errors and Ricci contractions in the proof of Proposition 3.1 .
Proposition 2.17. Fix n = 2 m ≥ 4 n=2m\ge4 n = 2 m ≥ 4 v , η > 0 v,\eta>0 v , η > 0 k = n − 2 k=n-2 k = n − 2 L 0 = L 0 ( n , v , η ) ≥ 32 L_0=L_0(n,v,\eta)\ge32 L 0 = L 0 ( n , v , η ) ≥ 32 L ≥ L 0 L\ge L_0 L ≥ L 0
N = B 2 R ( q ) ∖ B ‾ r x ( C ) \mathcal{N}=B_{2R}(q)\setminus\overline{B}_{r_x}(\mathcal{C}) N = B 2 R ( q ) ∖ B r x ( C ) be an ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) ( M n , g , J ) (M^n,g,J) ( M n , g , J ) n n n δ 1 > 0 \delta_1>0 δ 1 > 0
Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q ) ) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) , δ 1 R ≤ 4 δ 0 ≤ 1 / 100. \operatorname{Ric}_g\ge-(n-1)\delta_1^2g,\qquad\operatorname{Vol}(B_\rho(q))\ge v\rho^n\quad(0<\rho\le\delta_1^{-1}),\qquad\delta_1R\le4\delta_0\le1/100. Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q )) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) , δ 1 R ≤ 4 δ 0 ≤ 1/100. Let μ = μ N \mu=\mu_{\mathcal{N}} μ = μ N W δ 1 W^{\delta_1} W δ 1 (9) , with one fixed cutoff based at q q q
y ∈ N ∩ B R ( q ) , z ∈ C , r = d ( y , C ) = d ( y , z ) > 0 , s = L r ≤ 10 − 6 R , y\in\mathcal{N}\cap B_R(q),\qquad z\in\mathcal{C},\qquad r=d(y,\mathcal{C})=d(y,z)>0,\qquad s=Lr\le10^{-6}R, y ∈ N ∩ B R ( q ) , z ∈ C , r = d ( y , C ) = d ( y , z ) > 0 , s = L r ≤ 1 0 − 6 R , and put κ = ( n − 1 ) δ 1 2 \kappa=(n-1)\delta_1^2 κ = ( n − 1 ) δ 1 2 P P P (16) . There are ε 0 > 0 \varepsilon_0>0 ε 0 > 0 C < ∞ C<\infty C < ∞ 0 < c ≤ C < ∞ 0<c\le C<\infty 0 < c ≤ C < ∞ n , v , η , L n,v,\eta,L n , v , η , L P + κ r 2 + δ 0 ≤ ε 0 P+\kappa r^2+\delta_0\le\varepsilon_0 P + κ r 2 + δ 0 ≤ ε 0 u 1 , … , u k , h u^1,\ldots,u^k,h u 1 , … , u k , h B 8 s ( z ) B_{8s}(z) B 8 s ( z )
Δ u a = 0 , Δ h = 2 n , ∣ u ∣ ≤ C s , ∣ d u ∣ ≤ C , ∣ h ∣ ≤ C s 2 , ∣ d h ∣ ≤ C s , \Delta u^a=0,\qquad\Delta h=2n,\qquad|u|\le Cs,\qquad|du|\le C,\qquad|h|\le Cs^2,\qquad|dh|\le Cs, Δ u a = 0 , Δ h = 2 n , ∣ u ∣ ≤ C s , ∣ d u ∣ ≤ C , ∣ h ∣ ≤ C s 2 , ∣ d h ∣ ≤ C s , and
E : = r − n ∫ B r ( y ) [ ∑ a , b = 1 k ∣ ⟨ d u a , d u b ⟩ − δ a b ∣ 2 + ∑ j = 1 m − 1 ∣ ∂ ‾ w j ′ ∣ 2 + r 2 ∑ a = 1 k ( ∣ Hess u a ∣ 2 + ( Ric + κ g ) ( ∇ u a , ∇ u a ) ) + r − 2 ∑ a = 1 k ( ⟨ d h , d u a ⟩ − 2 u a ) 2 + ∣ Hess h − 2 g ∣ 2 + ( Ric + κ g ) ( ∇ h , ∇ h ) + r − 4 ( ∣ d h ∣ 2 − 4 h ) 2 ] d V g ≤ C ( P + κ r 2 ) . (27) \begin{aligned}
E:=r^{-n}\int_{B_r(y)}\bigg[&
\sum_{a,b=1}^{k}\left|\langle du^a,du^b\rangle-\delta_{ab}\right|^2+\sum_{j=1}^{m-1}|\overline{\partial}w'_j|^2\\
&+r^2\sum_{a=1}^{k}\left(|\operatorname{Hess}u^a|^2+(\operatorname{Ric}+\kappa g)(\nabla u^a,\nabla u^a)\right)\\
&+r^{-2}\sum_{a=1}^{k}\left(\langle dh,du^a\rangle-2u^a\right)^2\\
&+|\operatorname{Hess}h-2g|^2+(\operatorname{Ric}+\kappa g)(\nabla h,\nabla h)\\
&+r^{-4}\left(|dh|^2-4h\right)^2\bigg]\,dV_g\le C(P+\kappa r^2). \tag*{(27)}
\end{aligned} E := r − n ∫ B r ( y ) [ a , b = 1 ∑ k ⟨ d u a , d u b ⟩ − δ ab 2 + j = 1 ∑ m − 1 ∣ ∂ w j ′ ∣ 2 + r 2 a = 1 ∑ k ( ∣ Hess u a ∣ 2 + ( Ric + κ g ) ( ∇ u a , ∇ u a ) ) + r − 2 a = 1 ∑ k ( ⟨ d h , d u a ⟩ − 2 u a ) 2 + ∣ Hess h − 2 g ∣ 2 + ( Ric + κ g ) ( ∇ h , ∇ h ) + r − 4 ( ∣ d h ∣ 2 − 4 h ) 2 ] d V g ≤ C ( P + κ r 2 ) . (27) For b = h − ∣ u ∣ 2 b=h-|u|^2 b = h − ∣ u ∣ 2
c s 2 ≤ b ≤ C s 2 on B r / 8 ( y ) . cs^2\le b\le Cs^2\qquad\text{on }B_{r/8}(y). c s 2 ≤ b ≤ C s 2 on B r /8 ( y ) . Proof. By Corollary 2.12, with s = L r s=Lr s = L r b b b (27) other than the ∂ ‾ \overline{\partial} ∂ C P CP C P s s s u , h u,h u , h s − 1 u , s − 2 h s^{-1}u,s^{-2}h s − 1 u , s − 2 h Lemma 2.16 in this normalization. Indeed, the normalized stopping radius satisfies r z / s ≤ r / s = L − 1 ≤ 1 / 32 r_z/s\le r/s=L^{-1}\le1/32 r z / s ≤ r / s = L − 1 ≤ 1/32 B 16 s ( z ) ⊂ B 2 R ( q ) B_{16s}(z)\subset B_{2R}(q) B 16 s ( z ) ⊂ B 2 R ( q ) q q q s − n Vol ( B s ( z ) ) ≥ c ( n ) v s^{-n}\operatorname{Vol}(B_s(z)) \ge c(n)v s − n Vol ( B s ( z )) ≥ c ( n ) v C 0 = C 0 ( n , v , η , L ) C_0=C_0(n,v,\eta,L) C 0 = C 0 ( n , v , η , L )
F ≤ C P , ε = κ s 2 = L 2 κ r 2 . F \le CP,\qquad\varepsilon=\kappa s^2=L^2\kappa r^2. F ≤ C P , ε = κ s 2 = L 2 κ r 2 . Thus the hypotheses of that lemma hold after decreasing ε 0 ( n , v , η , L ) \varepsilon_0(n,v,\eta,L) ε 0 ( n , v , η , L ) B r ( y ) ⊂ B s ( z ) B_r(y)\subset B_s(z) B r ( y ) ⊂ B s ( z )
r − n ∫ B r ( y ) ∑ j = 1 m − 1 ∣ ∂ ‾ w j ′ ∣ 2 d V g ≤ C ( s r ) n ( F + κ s 2 ) ≤ C ( P + κ r 2 ) . r^{-n}\int_{B_r(y)}\sum_{j=1}^{m-1}\lvert\overline{\partial}w'_j\rvert^2\,dV_g\le C\left(\frac{s}{r}\right)^n(F+\kappa s^2)\le C(P+\kappa r^2). r − n ∫ B r ( y ) j = 1 ∑ m − 1 ∣ ∂ w j ′ ∣ 2 d V g ≤ C ( r s ) n ( F + κ s 2 ) ≤ C ( P + κ r 2 ) . The orthogonal change preserves ∣ u ∣ 2 \lvert u\rvert^2 ∣ u ∣ 2 b b b
A local Ricci estimate in terms of the energy
This section proves a Ricci L 1 L^1 L 1 B r ( y ) B_r(y) B r ( y ) B 8 r ( y ) B_{8r}(y) B 8 r ( y ) Proposition 3.1 gives a linear bound in their energy E E E Corollary 2.15 and Proposition 2.17 provide these data on regular balls in a neck; we make this application in § 4 .
Let B 8 r ( y ) B_{8r}(y) B 8 r ( y ) ( M n , g , J ) (M^n,g,J) ( M n , g , J ) n = 2 m ≥ 4 n=2m\ge4 n = 2 m ≥ 4 C 0 ≥ 1 C_0\ge1 C 0 ≥ 1 κ ≥ 0 \kappa\ge0 κ ≥ 0
Ric g ≥ − κ g on B 8 r ( y ) , κ r 2 ≤ 1. \operatorname{Ric}_g\ge-\kappa g\quad\text{on }B_{8r}(y),\qquad\kappa r^2\le1. Ric g ≥ − κ g on B 8 r ( y ) , κ r 2 ≤ 1. Suppose that there is a holomorphic chart z : B 8 r ( y ) ⟶ C m z:B_{8r}(y)\longrightarrow\mathbb{C}^m z : B 8 r ( y ) ⟶ C m z ( y ) = 0 z(y)=0 z ( y ) = 0
∣ z ( x ) − z ( x ′ ) ∣ − d g ( x , x ′ ) ≤ 10 − 3 r ( x , x ′ ∈ B 8 r ( y ) ) , \bigl\lvert z(x)-z(x')\bigr\rvert-d_g(x,x')\le10^{-3}r\qquad(x,x'\in B_{8r}(y)), z ( x ) − z ( x ′ ) − d g ( x , x ′ ) ≤ 1 0 − 3 r ( x , x ′ ∈ B 8 r ( y )) , and
g ≥ C 0 − 1 z ∗ g C m , ω = i ∂ ∂ ‾ ρ , ∣ ρ ∣ ≤ C 0 r 2 . g\ge C_0^{-1}z^*g_{\mathbb{C}^m},\qquad\omega=i\partial\overline{\partial}\rho,\qquad\lvert\rho\rvert\le C_0r^2. g ≥ C 0 − 1 z ∗ g C m , ω = i ∂ ∂ ρ , ∣ ρ ∣ ≤ C 0 r 2 . We use the Kähler and coordinate conventions of § 2.5 . Let u 1 , … , u n − 2 , h u^1,\ldots,u^{n-2},h u 1 , … , u n − 2 , h B 8 r ( y ) B_{8r}(y) B 8 r ( y )
Δ u a = 0 , Δ h = 2 n , b = h − ∣ u ∣ 2 , \Delta u^a=0,\qquad\Delta h=2n,\qquad b=h-\lvert u\rvert^2, Δ u a = 0 , Δ h = 2 n , b = h − ∣ u ∣ 2 , and
C 0 − 1 r 2 ≤ b ≤ C 0 r 2 , r − 1 ∣ u ∣ + ∣ d u ∣ + r − 2 ∣ h ∣ + r − 1 ∣ d h ∣ ≤ C 0 . C_0^{-1}r^2\le b\le C_0r^2,\qquad r^{-1}\lvert u\rvert+\lvert du\rvert+r^{-2}\lvert h\rvert+r^{-1}\lvert dh\rvert\le C_0. C 0 − 1 r 2 ≤ b ≤ C 0 r 2 , r − 1 ∣ u ∣ + ∣ d u ∣ + r − 2 ∣ h ∣ + r − 1 ∣ d h ∣ ≤ C 0 . For the paired complex functions, we keep the notation w j ′ w'_j w j ′ Lemma 2.16 . Define
E : = r − n ∫ B 8 r ( y ) [ ∑ a , b = 1 n − 2 ∣ ⟨ d u a , d u b ⟩ − δ a b ∣ 2 + ∑ j = 1 m − 1 ∣ ∂ ‾ w j ′ ∣ 2 + r 2 ∑ a = 1 n − 2 ( ∣ Hess u a ∣ 2 + ( Ric + κ g ) ( ∇ u a , ∇ u a ) ) + r − 2 ∑ a = 1 n − 2 ( ⟨ d h , d u a ⟩ − 2 u a ) 2 + ∣ Hess h − 2 g ∣ 2 + ( Ric + κ g ) ( ∇ h , ∇ h ) + r − 4 ( ∣ d h ∣ 2 − 4 h ) 2 ] d V g . (28) \begin{aligned}
E:=r^{-n}\int_{B_{8r}(y)}\biggl[&
\sum_{a,b=1}^{n-2}\bigl\lvert\langle du^a,du^b\rangle-\delta_{ab}\bigr\rvert^2+\sum_{j=1}^{m-1}\lvert\overline{\partial}w'_j\rvert^2\\
&+r^2\sum_{a=1}^{n-2}\left(\lvert\operatorname{Hess}u^a\rvert^2+(\operatorname{Ric}+\kappa g)(\nabla u^a,\nabla u^a)\right)\\
&+r^{-2}\sum_{a=1}^{n-2}\bigl(\langle dh,du^a\rangle-2u^a\bigr)^2\\
&+\lvert\operatorname{Hess}h-2g\rvert^2+(\operatorname{Ric}+\kappa g)(\nabla h,\nabla h)\\
&+r^{-4}\bigl(\lvert dh\rvert^2-4h\bigr)^2\biggr]\,dV_g. \tag*{(28)}
\end{aligned} E := r − n ∫ B 8 r ( y ) [ a , b = 1 ∑ n − 2 ⟨ d u a , d u b ⟩ − δ ab 2 + j = 1 ∑ m − 1 ∣ ∂ w j ′ ∣ 2 + r 2 a = 1 ∑ n − 2 ( ∣ Hess u a ∣ 2 + ( Ric + κ g ) ( ∇ u a , ∇ u a ) ) + r − 2 a = 1 ∑ n − 2 ( ⟨ d h , d u a ⟩ − 2 u a ) 2 + ∣ Hess h − 2 g ∣ 2 + ( Ric + κ g ) ( ∇ h , ∇ h ) + r − 4 ( ∣ d h ∣ 2 − 4 h ) 2 ] d V g . (28) In particular, E E E χ \chi χ
0 ≤ χ ≤ 1 , χ = 1 on B 2 r ( y ) , supp χ ⊂ B 3 r ( y ) , 0 \le\chi\le1,\qquad\chi=1\text{ on }B_{2r}(y),\qquad\operatorname{supp}\chi\subset B_{3r}(y), 0 ≤ χ ≤ 1 , χ = 1 on B 2 r ( y ) , supp χ ⊂ B 3 r ( y ) , ∣ i ∂ ∂ ˉ χ ∣ 0 ≤ C ( n ) r − 2 . \lvert i\partial\bar{\partial}\chi\rvert_0\le C(n)r^{-2}. ∣ i ∂ ∂ ˉ χ ∣ 0 ≤ C ( n ) r − 2 . The coordinate distortion gives such a cutoff by rescaling a fixed smooth radial profile. The following proposition bounds the Ricci integral by this local energy, with the error from the Ricci lower bound written at scale r r r
Proposition 3.1. In the local setting above, there is C = C ( n , C 0 ) < ∞ C=C(n,C_0)<\infty C = C ( n , C 0 ) < ∞
∫ B r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ( E + κ r 2 ) . (29) \int_{B_r(y)}\lvert\operatorname{Ric}\rvert\,dV_g\le Cr^{n-2}(E+\kappa r^2). \tag*{(29)} ∫ B r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ( E + κ r 2 ) . (29) Idea of the proof. We first explain the proof for E ≤ E 0 ( n , C 0 ) E\le E_0(n,C_0) E ≤ E 0 ( n , C 0 ) E E E
s 0 = 2 − m / 2 ∂ w 1 ′ ∧ ⋯ ∧ ∂ w m − 1 ′ ∧ ∂ log b . (30) s_0=2^{-m/2}\partial w'_1\wedge\cdots\wedge\partial w'_{m-1}\wedge\partial\log b. \tag*{(30)} s 0 = 2 − m /2 ∂ w 1 ′ ∧ ⋯ ∧ ∂ w m − 1 ′ ∧ ∂ log b . (30) The product b ∣ s 0 ∣ 2 b\lvert s_0\rvert^2 b ∣ s 0 ∣ 2 1 1 1
S + n κ = ( S + n κ ) b ∣ s 0 ∣ 2 + ( S + n κ ) ( 1 − b ∣ s 0 ∣ 2 ) . S+n\kappa=(S+n\kappa)b\lvert s_0\rvert^2+(S+n\kappa)(1-b\lvert s_0\rvert^2). S + nκ = ( S + nκ ) b ∣ s 0 ∣ 2 + ( S + nκ ) ( 1 − b ∣ s 0 ∣ 2 ) . The first term. The linear algebra in Lemma 3.2 gives
∫ ( S + n κ ) b ∣ s 0 ∣ 2 d V g ≤ C ∫ [ ∑ a ( Ric + κ g ) ( ∇ u a , ∇ u a ) + r − 2 ( Ric + κ g ) ( ∇ h , ∇ h ) ] d V g ≤ C r n − 2 E . \int(S+n\kappa)b\lvert s_0\rvert^2\,dV_g
\le C\int\left[\sum_a(\operatorname{Ric}+\kappa g)(\nabla u^a,\nabla u^a)
+r^{-2}(\operatorname{Ric}+\kappa g)(\nabla h,\nabla h)\right]dV_g
\le Cr^{n-2}E. ∫ ( S + nκ ) b ∣ s 0 ∣ 2 d V g ≤ C ∫ [ a ∑ ( Ric + κ g ) ( ∇ u a , ∇ u a ) + r − 2 ( Ric + κ g ) ( ∇ h , ∇ h ) ] d V g ≤ C r n − 2 E . The radial term Ric ( ∇ h , ∇ h ) \operatorname{Ric}(\nabla h,\nabla h) Ric ( ∇ h , ∇ h ) h h h u a u^a u a E E E r r r
The second term. This is the main difficulty, but its factor 1 − b ∣ s 0 ∣ 2 1-b\lvert s_0\rvert^2 1 − b ∣ s 0 ∣ 2 L 2 L^2 L 2 E \sqrt{E} E
r − n ∫ ( 1 − b ∣ s 0 ∣ 2 ) 2 d V g ≤ C E . r^{-n}\int(1-b\lvert s_0\rvert^2)^2\,dV_g\le CE. r − n ∫ ( 1 − b ∣ s 0 ∣ 2 ) 2 d V g ≤ C E . Indeed, the squared norm is a Gram determinant, and E E E s 0 s_0 s 0 Lemma 3.4 gives such a replacement and bounds the curvature-weighted change by C r n − 2 E Cr^{n-2}E C r n − 2 E s 0 s_0 s 0 s s s
Positivity and the bounded Kähler potential first reduce the scalar estimate to a coordinate cutoff integral, as in (52) . Holomorphicity gives
2 π Ric ( ω ) + κ ω = i ∂ ∂ ˉ log ( b ∣ s 0 ∣ 2 ) − i ∂ ∂ ˉ log b + κ ω . 2\pi\operatorname{Ric}(\omega)+\kappa\omega=i\partial\bar{\partial}\log(b\lvert s_0\rvert^2)-i\partial\bar{\partial}\log b+\kappa\omega. 2 π Ric ( ω ) + κω = i ∂ ∂ ˉ log ( b ∣ s 0 ∣ 2 ) − i ∂ ∂ ˉ log b + κω . To estimate the remaining signed integral, we use the identity
( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + κ ω ) = ( b ∣ s 0 ∣ 2 − 1 ) i ∂ ∂ ˉ log b + i b ∣ s 0 ∣ 2 ∂ log ( b ∣ s 0 ∣ 2 ) ∧ ∂ ˉ log ( b ∣ s 0 ∣ 2 ) + κ ( 1 − b ∣ s 0 ∣ 2 ) ω − i ∂ ∂ ˉ [ b ∣ s 0 ∣ 2 − 1 − log ( b ∣ s 0 ∣ 2 ) ] . \begin{aligned}
(1-b|s_0|^2)(2\pi\operatorname{Ric}(\omega)+\kappa\omega)
={}&(b|s_0|^2-1)i\partial\bar{\partial}\log b\\
&+ib|s_0|^2\partial\log(b|s_0|^2)\wedge\bar{\partial}\log(b|s_0|^2)\\
&+\kappa(1-b|s_0|^2)\omega\\
&-i\partial\bar{\partial}[b|s_0|^2-1-\log(b|s_0|^2)].
\end{aligned} ( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + κω ) = ( b ∣ s 0 ∣ 2 − 1 ) i ∂ ∂ ˉ log b + ib ∣ s 0 ∣ 2 ∂ log ( b ∣ s 0 ∣ 2 ) ∧ ∂ ˉ log ( b ∣ s 0 ∣ 2 ) + κ ( 1 − b ∣ s 0 ∣ 2 ) ω − i ∂ ∂ ˉ [ b ∣ s 0 ∣ 2 − 1 − log ( b ∣ s 0 ∣ 2 )] . We integrate against χ ω 0 m − 1 \chi\omega_0^{m-1} χ ω 0 m − 1 r r r ( 1 , 1 ) (1,1) ( 1 , 1 ) log b \log b log b
∥ 1 − b ∣ s 0 ∣ 2 ∥ L 2 ∥ i ∂ ∂ ˉ log b ∥ L 2 ≤ C ( r n / 2 E ) ( r n / 2 − 2 E ) = C r n − 2 E . \|1-b|s_0|^2\|_{L^2}\|i\partial\bar{\partial}\log b\|_{L^2}
\leq C\left(r^{n/2}\sqrt{E}\right)\left(r^{n/2-2}\sqrt{E}\right)=Cr^{n-2}E. ∥1 − b ∣ s 0 ∣ 2 ∥ L 2 ∥ i ∂ ∂ ˉ log b ∥ L 2 ≤ C ( r n /2 E ) ( r n /2 − 2 E ) = C r n − 2 E . The second term is a squared covariant derivative. For the original section, the product rule uses the cancellation
D ′ ∂ log b + ∂ log b ⊗ ∂ log b = b − 1 ( Hess b ) 2 , 0 . D'\partial\log b+\partial\log b\otimes\partial\log b=b^{-1}(\operatorname{Hess}b)^{2,0}. D ′ ∂ log b + ∂ log b ⊗ ∂ log b = b − 1 ( Hess b ) 2 , 0 . This vanishes on the model and is controlled by the Hessian and complex-pairing errors. The holomorphic correction preserves this estimate. The third term contains κ r 2 \kappa r^2 κ r 2
∫ b ∣ s 0 ∣ 2 ∣ ∂ log ( b ∣ s 0 ∣ 2 ) ∣ 2 d V g = ∫ b ∣ D ′ s 0 + ∂ log b ⊗ s 0 ∣ 2 d V g ≤ C r n − 2 E , \int b|s_0|^2|\partial\log(b|s_0|^2)|^2\,dV_g
=\int b|D's_0+\partial\log b\otimes s_0|^2\,dV_g
\leq Cr^{n-2}E, ∫ b ∣ s 0 ∣ 2 ∣ ∂ log ( b ∣ s 0 ∣ 2 ) ∣ 2 d V g = ∫ b ∣ D ′ s 0 + ∂ log b ⊗ s 0 ∣ 2 d V g ≤ C r n − 2 E , κ ∫ ∣ 1 − b ∣ s 0 ∣ 2 ∣ ∣ d V g ≤ C κ r n E ≤ C r n − 2 ( E + κ 2 r 4 ) . \kappa\int|1-b|s_0|^2||\,dV_g\leq C\kappa r^n\sqrt{E}\leq Cr^{n-2}(E+\kappa^2r^4). κ ∫ ∣1 − b ∣ s 0 ∣ 2 ∣∣ d V g ≤ C κ r n E ≤ C r n − 2 ( E + κ 2 r 4 ) . The metric lower bound gives the same bounds for the coordinate cutoff integrals. For the last term, integrate by parts twice. The derivatives fall on the cutoff, and the integral is with respect to d V 0 dV_0 d V 0 s 0 = 2 − m / 2 a d z 1 ∧ ⋯ ∧ d z m s_0=2^{-m/2}a\,dz_1\wedge\cdots\wedge dz_m s 0 = 2 − m /2 a d z 1 ∧ ⋯ ∧ d z m (48) and Lemma 3.4 give
d V 0 = b ∣ s 0 ∣ 2 b ∣ a ∣ 2 d V g , b ∣ a ∣ 2 ≥ c . dV_0=\frac{b|s_0|^2}{b|a|^2}\,dV_g,\qquad b|a|^2\geq c. d V 0 = b ∣ a ∣ 2 b ∣ s 0 ∣ 2 d V g , b ∣ a ∣ 2 ≥ c . Thus the factor t = b ∣ s 0 ∣ 2 t=b|s_0|^2 t = b ∣ s 0 ∣ 2 0 ≤ t ( t − 1 − log t ) ≤ ( t − 1 ) 2 0\leq t(t-1-\log t)\leq(t-1)^2 0 ≤ t ( t − 1 − log t ) ≤ ( t − 1 ) 2 r − 2 r^{-2} r − 2
C r − 2 ∫ [ b ∣ s 0 ∣ 2 − 1 − log ( b ∣ s 0 ∣ 2 ) ] d V 0 ≤ C r − 2 ∫ ( b ∣ s 0 ∣ 2 − 1 ) 2 d V g ≤ C r n − 2 E . Cr^{-2}\int[b|s_0|^2-1-\log(b|s_0|^2)]\,dV_0
\leq Cr^{-2}\int(b|s_0|^2-1)^2\,dV_g
\leq Cr^{n-2}E. C r − 2 ∫ [ b ∣ s 0 ∣ 2 − 1 − log ( b ∣ s 0 ∣ 2 )] d V 0 ≤ C r − 2 ∫ ( b ∣ s 0 ∣ 2 − 1 ) 2 d V g ≤ C r n − 2 E . Lemma 3.3 estimates this remaining integral. The final proof of Proposition 3.1 combines it with Lemma 3.2 and the positive-form comparison to obtain, since κ r 2 ≤ 1 \kappa r^2\leq1 κ r 2 ≤ 1
∫ B r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ( E + κ r 2 ) . \int_{B_r(y)}|\operatorname{Ric}|\,dV_g\leq Cr^{n-2}(E+\kappa r^2). ∫ B r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ( E + κ r 2 ) . The weighted term
We begin with the weighted scalar-curvature term, for which linear algebra reduces the problem to Ricci contractions with ∇ u a \nabla u^{a} ∇ u a ∇ h \nabla h ∇ h E E E
Lemma 3.2. In the local setting above, one has
∫ B 8 r ( y ) ( S + n κ ) b ∣ s 0 ∣ 2 d V g ≤ C ∫ B 8 r ( y ) [ ∑ a = 1 n − 2 ( Ric + κ g ) ( ∇ u a , ∇ u a ) + r − 2 ( Ric + κ g ) ( ∇ h , ∇ h ) ] d V g , (31) \int_{B_{8r}(y)} (S+n\kappa)b|s_{0}|^{2}\,dV_{g}
\leq C\int_{B_{8r}(y)}\left[\sum_{a=1}^{n-2}(\operatorname{Ric}+\kappa g)(\nabla u^{a},\nabla u^{a})+r^{-2}(\operatorname{Ric}+\kappa g)(\nabla h,\nabla h)\right]\,dV_{g}, \tag*{(31)} ∫ B 8 r ( y ) ( S + nκ ) b ∣ s 0 ∣ 2 d V g ≤ C ∫ B 8 r ( y ) [ a = 1 ∑ n − 2 ( Ric + κ g ) ( ∇ u a , ∇ u a ) + r − 2 ( Ric + κ g ) ( ∇ h , ∇ h ) ] d V g , (31) where C = C ( n , C 0 ) C=C(n,C_{0}) C = C ( n , C 0 )
Proof. Rescale g , u , h g,u,h g , u , h r − 2 g , r − 1 u , r − 2 h r^{-2}g,r^{-1}u,r^{-2}h r − 2 g , r − 1 u , r − 2 h ε = κ r 2 \varepsilon=\kappa r^{2} ε = κ r 2 r = 1 r=1 r = 1
At a fixed point, choose an orthonormal eigenbasis e 1 , J e 1 , … , e m , J e m e_{1},Je_{1},\ldots,e_{m},Je_{m} e 1 , J e 1 , … , e m , J e m Ric + ε g \operatorname{Ric}+\varepsilon g Ric + ε g r 1 , r 1 , … , r m , r m ≥ 0 r_{1},r_{1},\ldots,r_{m},r_{m}\geq0 r 1 , r 1 , … , r m , r m ≥ 0 η 1 , … , η m \eta^{1},\ldots,\eta^{m} η 1 , … , η m ( 1 , 0 ) (1,0) ( 1 , 0 )
e ℓ − i J e ℓ 2 , 1 ≤ ℓ ≤ m . \frac{e_{\ell}-iJe_{\ell}}{\sqrt{2}},\qquad1\leq\ell\leq m. 2 e ℓ − i J e ℓ , 1 ≤ ℓ ≤ m . Write
θ j = 2 − 1 / 2 ∂ w j ′ ( j < m ) , θ m = 2 − 1 / 2 ∂ log b , θ j = ∑ ℓ a j ℓ η ℓ . \theta_{j}=2^{-1/2}\partial w'_{j}\quad(j<m),\qquad\theta_{m}=2^{-1/2}\partial\log b,\qquad\theta_{j}=\sum_{\ell}a_{j\ell}\eta^{\ell}. θ j = 2 − 1/2 ∂ w j ′ ( j < m ) , θ m = 2 − 1/2 ∂ log b , θ j = ℓ ∑ a j ℓ η ℓ . Then s 0 = θ 1 ∧ ⋯ ∧ θ m s_{0}=\theta_{1}\wedge\cdots\wedge\theta_{m} s 0 = θ 1 ∧ ⋯ ∧ θ m
A ℓ q = ∑ j a j ℓ ‾ a j q , R = diag ( r 1 , … , r m ) , A_{\ell q}=\sum_{j}\overline{a_{j\ell}}a_{jq},\qquad R=\operatorname{diag}(r_{1},\ldots,r_{m}), A ℓ q = j ∑ a j ℓ a j q , R = diag ( r 1 , … , r m ) , we have
det A = ∣ s 0 ∣ 2 , tr R = 1 2 ( S + n ε ) , tr ( A R ) = ∑ j , ℓ r ℓ ∣ a j ℓ ∣ 2 . \det A=|s_{0}|^{2},\qquad\operatorname{tr}R=\frac{1}{2}(S+n\varepsilon),\qquad\operatorname{tr}(AR)=\sum_{j,\ell}r_{\ell}|a_{j\ell}|^{2}. det A = ∣ s 0 ∣ 2 , tr R = 2 1 ( S + n ε ) , tr ( A R ) = j , ℓ ∑ r ℓ ∣ a j ℓ ∣ 2 . Since
d log b = b − 1 ( d h − 2 ∑ a u a d u a ) , d\log b=b^{-1}\left(dh-2\sum_{a}u^{a}\,du^{a}\right), d log b = b − 1 ( d h − 2 a ∑ u a d u a ) , the assumptions give 0 ≤ A ≤ C Id m 0\leq A\leq C\operatorname{Id}_{m} 0 ≤ A ≤ C Id m A A A λ i ∈ [ 0 , C ] \lambda_{i}\in[0,C] λ i ∈ [ 0 , C ] R R R
det A tr R = ∑ i ( ∏ j ≠ i λ j ) λ i R i ˉ i ≤ C m − 1 ∑ i λ i R i ˉ i = C m − 1 tr ( A R ) . (32) \det A\operatorname{tr}R=\sum_{i}\left(\prod_{j\neq i}\lambda_{j}\right)\lambda_{i}R_{\bar{i}i}\leq C^{m-1}\sum_{i}\lambda_{i}R_{\bar{i}i}=C^{m-1}\operatorname{tr}(AR). \tag*{(32)} det A tr R = i ∑ j = i ∏ λ j λ i R i ˉ i ≤ C m − 1 i ∑ λ i R i ˉ i = C m − 1 tr ( A R ) . (32) This also holds when A A A
The unitary normalization gives the contraction identities
∑ ℓ r ℓ ∣ a j ℓ ∣ 2 = 1 4 ( Ric + ε g ) ( ∇ u 2 j − 1 − J ∇ u 2 j , ∇ u 2 j − 1 − J ∇ u 2 j ) ( j < m ) , \sum_{\ell}r_{\ell}|a_{j\ell}|^{2}=\frac{1}{4}(\operatorname{Ric}+\varepsilon g)(\nabla u^{2j-1}-J\nabla u^{2j},\nabla u^{2j-1}-J\nabla u^{2j})\qquad(j<m), ℓ ∑ r ℓ ∣ a j ℓ ∣ 2 = 4 1 ( Ric + ε g ) ( ∇ u 2 j − 1 − J ∇ u 2 j , ∇ u 2 j − 1 − J ∇ u 2 j ) ( j < m ) , ∑ ℓ r ℓ ∣ a m ℓ ∣ 2 = 1 4 ( Ric + ε g ) ( ∇ log b , ∇ log b ) . \sum_{\ell}r_{\ell}|a_{m\ell}|^{2}=\frac{1}{4}(\operatorname{Ric}+\varepsilon g)(\nabla\log b,\nabla\log b). ℓ ∑ r ℓ ∣ a m ℓ ∣ 2 = 4 1 ( Ric + ε g ) ( ∇ log b , ∇ log b ) . By positivity and J J J
( S + n ε ) ∣ s 0 ∣ 2 ≤ C [ ∑ a ( Ric + ε g ) ( ∇ u a , ∇ u a ) + ( Ric + ε g ) ( ∇ log b , ∇ log b ) ] . (S+n\varepsilon)|s_{0}|^{2}\leq C\left[\sum_{a}(\operatorname{Ric}+\varepsilon g)(\nabla u^{a},\nabla u^{a})+(\operatorname{Ric}+\varepsilon g)(\nabla\log b,\nabla\log b)\right]. ( S + n ε ) ∣ s 0 ∣ 2 ≤ C [ a ∑ ( Ric + ε g ) ( ∇ u a , ∇ u a ) + ( Ric + ε g ) ( ∇ log b , ∇ log b ) ] . Finally, by Cauchy–Schwarz for the nonnegative form Ric + ε g \operatorname{Ric}+\varepsilon g Ric + ε g
( Ric + ε g ) ( ∇ log b , ∇ log b ) ≤ b − 2 ( 1 + 4 ∣ u ∣ 2 ) [ ( Ric + ε g ) ( ∇ h , ∇ h ) + ∑ a ( Ric + ε g ) ( ∇ u a , ∇ u a ) ] . (\operatorname{Ric}+\varepsilon g)(\nabla\log b,\nabla\log b)
\le b^{-2}(1+4|u|^2)\left[(\operatorname{Ric}+\varepsilon g)(\nabla h,\nabla h)+\sum_a(\operatorname{Ric}+\varepsilon g)(\nabla u^a,\nabla u^a)\right]. ( Ric + ε g ) ( ∇ log b , ∇ log b ) ≤ b − 2 ( 1 + 4∣ u ∣ 2 ) [ ( Ric + ε g ) ( ∇ h , ∇ h ) + a ∑ ( Ric + ε g ) ( ∇ u a , ∇ u a ) ] . Multiplying the pointwise bound by b ≤ C b\le C b ≤ C (31) .
Remark. The same estimate holds on any Kähler domain U U U U U U B 8 r ( y ) B_{8r}(y) B 8 r ( y ) u : U → R n − 2 u:U\to\mathbb{R}^{n-2} u : U → R n − 2 h : U → R h:U\to\mathbb{R} h : U → R
Ric g ≥ − κ g , b = h − ∣ u ∣ 2 , c r 2 ≤ b ≤ C 0 r 2 , \operatorname{Ric}_g\ge-\kappa g,\qquad b=h-|u|^2,\qquad cr^2\le b\le C_0r^2, Ric g ≥ − κ g , b = h − ∣ u ∣ 2 , c r 2 ≤ b ≤ C 0 r 2 , r − 1 ∣ u ∣ + ∣ d u ∣ + r − 1 ∣ d h ∣ ≤ C 0 , r^{-1}|u|+|du|+r^{-1}|dh|\le C_0, r − 1 ∣ u ∣ + ∣ d u ∣ + r − 1 ∣ d h ∣ ≤ C 0 , where r > 0 r>0 r > 0 κ ≥ 0 \kappa\ge0 κ ≥ 0 c > 0 c>0 c > 0 s 0 s_0 s 0 (30) , and the constant depends only on n , c , C 0 n,c,C_0 n , c , C 0
The remaining term
We now estimate the remaining term, whose small factor 1 − b ∣ s 0 ∣ 2 1-b|s_0|^2 1 − b ∣ s 0 ∣ 2 E E E Lemma 3.3 gives the required signed integral estimate, and Lemma 3.4 provides the holomorphic replacement needed to prove it.
The next lemma controls the signed cutoff integral of the remaining term at scale r r r E E E
Lemma 3.3. Use the local setting and the cutoff fixed above. There are E 0 > 0 E_0>0 E 0 > 0 C < ∞ C<\infty C < ∞ n , C 0 n,C_0 n , C 0 E ≤ E 0 E\le E_0 E ≤ E 0
∣ ∫ B 3 r ( y ) χ ( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + κ ω ) ∧ ω 0 m − 1 ∣ ≤ C r n − 2 ( E + κ r 4 ) . (33) \left|\int_{B_{3r}(y)}\chi(1-b|s_0|^2)(2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega_0^{m-1}\right|\le Cr^{n-2}(E+\kappa r^4). \tag*{(33)} ∫ B 3 r ( y ) χ ( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + κω ) ∧ ω 0 m − 1 ≤ C r n − 2 ( E + κ r 4 ) . (33) To prove Lemma 3.3 , we need a holomorphic replacement for s 0 s_0 s 0 d V 0 dV_0 d V 0 d V g dV_g d V g
Lemma 3.4. Use the local setting fixed above. There are E 0 , c > 0 E_0,c>0 E 0 , c > 0 C < ∞ C<\infty C < ∞ n , C 0 n,C_0 n , C 0 E ≤ E 0 E\le E_0 E ≤ E 0
s = 2 − m / 2 a ( z ) d z 1 ∧ ⋯ ∧ d z m s=2^{-m/2}a(z)\,dz_1\wedge\cdots\wedge dz_m s = 2 − m /2 a ( z ) d z 1 ∧ ⋯ ∧ d z m of the canonical bundle on B 6 r ( y ) B_{6r}(y) B 6 r ( y )
∫ B 4 r ( y ) ( ( S + n κ ) b ∣ s ∣ 2 − ∣ s 0 ∣ 2 ) d V g ≤ C r n − 2 E , (34) \int_{B_{4r}(y)}\left((S+n\kappa)b|s|^2-|s_0|^2\right)\,dV_g\le Cr^{n-2}E, \tag*{(34)} ∫ B 4 r ( y ) ( ( S + nκ ) b ∣ s ∣ 2 − ∣ s 0 ∣ 2 ) d V g ≤ C r n − 2 E , (34) ∫ B 4 r ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≤ C r n E , (35) \int_{B_{4r}(y)}(b|s|^2-1)^2\,dV_g\le Cr^nE, \tag*{(35)} ∫ B 4 r ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≤ C r n E , (35) and
∫ B 4 r ( y ) ( r 2 ∣ i ∂ ∂ ˉ log b ∣ 2 + b ∣ D ′ s + ∂ log b ⊗ s ∣ 2 ) d V g ≤ C r n − 2 E . (36) \int_{B_{4r}(y)}\left(r^2|i\partial\bar{\partial}\log b|^2+b|D's+\partial\log b\otimes s|^2\right)\,dV_g\le Cr^{n-2}E. \tag*{(36)} ∫ B 4 r ( y ) ( r 2 ∣ i ∂ ∂ ˉ log b ∣ 2 + b ∣ D ′ s + ∂ log b ⊗ s ∣ 2 ) d V g ≤ C r n − 2 E . (36) Moreover,
r ∣ a ∣ ≥ c on B 4 r ( y ) . (37) r|a|\ge c\qquad\text{on }B_{4r}(y). \tag*{(37)} r ∣ a ∣ ≥ c on B 4 r ( y ) . (37) Proof. Rescale g , u , h g,u,h g , u , h r − 2 g , r − 1 u , r − 2 h r^{-2}g,r^{-1}u,r^{-2}h r − 2 g , r − 1 u , r − 2 h r r r ε = κ r 2 ≤ 1 \varepsilon=\kappa r^2\le1 ε = κ r 2 ≤ 1 s 0 s_0 s 0 r − ( m − 1 ) r^{-(m-1)} r − ( m − 1 ) r r r b ∣ s 0 ∣ 2 b|s_0|^2 b ∣ s 0 ∣ 2 r = 1 r=1 r = 1 n , C 0 n,C_0 n , C 0 E E E Vol ( B 8 ( y ) ) ≤ C ( n ) \operatorname{Vol}(B_8(y))\le C(n) Vol ( B 8 ( y )) ≤ C ( n )
B 6 ( y ) ⊂ B 13 / 2 0 ( 0 ) ⊂ B 7 0 ( 0 ) ⊂ B 8 ( y ) . B_6(y)\subset B_{13/2}^0(0)\subset B_7^0(0)\subset B_8(y). B 6 ( y ) ⊂ B 13/2 0 ( 0 ) ⊂ B 7 0 ( 0 ) ⊂ B 8 ( y ) . Put ℓ a = ⟨ d h , d u a ⟩ − 2 u a \ell_a=\langle dh,du^a\rangle-2u^a ℓ a = ⟨ d h , d u a ⟩ − 2 u a b = h − ∣ u ∣ 2 b=h-|u|^2 b = h − ∣ u ∣ 2
∣ d b ∣ 2 − 4 b = ( ∣ d h ∣ 2 − 4 h ) + 4 ∑ a , b u a u b ( ⟨ d u a , d u b ⟩ − δ a b ) − 4 ∑ a u a ℓ a , ⟨ d b , d u a ⟩ = ℓ a − 2 ∑ b u b ( ⟨ d u a , d u b ⟩ − δ a b ) . (38) \begin{aligned}
|db|^2-4b&=(|dh|^2-4h)+4\sum_{a,b}u^a u^b\left(\langle du^a,du^b\rangle-\delta_{ab}\right)-4\sum_a u^a\ell_a,\\
\langle db,du^a\rangle&=\ell_a-2\sum_bu^b\left(\langle du^a,du^b\rangle-\delta_{ab}\right). \tag*{(38)}
\end{aligned} ∣ d b ∣ 2 − 4 b ⟨ d b , d u a ⟩ = ( ∣ d h ∣ 2 − 4 h ) + 4 a , b ∑ u a u b ( ⟨ d u a , d u b ⟩ − δ ab ) − 4 a ∑ u a ℓ a , = ℓ a − 2 b ∑ u b ( ⟨ d u a , d u b ⟩ − δ ab ) . (38) Together with the complex pairing term in E E E ∇ u a \nabla u^a ∇ u a ∇ b / ( 2 b ) \nabla b/(2\sqrt b) ∇ b / ( 2 b ) J ∇ b / ( 2 b ) J\nabla b/(2\sqrt b) J ∇ b / ( 2 b ) C E CE C E
∫ B 8 ( y ) ∣ g − ∑ a d u a ⊗ d u a − d b ⊗ d b + ( d b ∘ J ) ⊗ ( d b ∘ J ) 4 b ∣ 2 d V g ≤ C E . (39) \int_{B_8(y)}\left|g-\sum_a du^a\otimes du^a-\frac{db\otimes db+(db\circ J)\otimes(db\circ J)}{4b}\right|^2dV_g\le CE. \tag*{(39)} ∫ B 8 ( y ) g − a ∑ d u a ⊗ d u a − 4 b d b ⊗ d b + ( d b ∘ J ) ⊗ ( d b ∘ J ) 2 d V g ≤ C E . (39) The Hessian identity
b ( Hess log b + 1 2 [ d log b ⊗ d log b − ( d log b ∘ J ) ⊗ ( d log b ∘ J ) ] ) = ( Hess h − 2 g ) − 2 ∑ a u a Hess u a + 2 ( g − ∑ a d u a ⊗ d u a − d b ⊗ d b + ( d b ∘ J ) ⊗ ( d b ∘ J ) 4 b ) (40) \begin{aligned}
b\left(\operatorname{Hess}\log b+\frac12\left[d\log b\otimes d\log b-(d\log b\circ J)\otimes(d\log b\circ J)\right]\right)
={}&(\operatorname{Hess}h-2g)-2\sum_a u^a\operatorname{Hess}u^a\\
&+2\left(g-\sum_a du^a\otimes du^a-\frac{db\otimes db+(db\circ J)\otimes(db\circ J)}{4b}\right) \tag*{(40)}
\end{aligned} b ( Hess log b + 2 1 [ d log b ⊗ d log b − ( d log b ∘ J ) ⊗ ( d log b ∘ J ) ] ) = ( Hess h − 2 g ) − 2 a ∑ u a Hess u a + 2 ( g − a ∑ d u a ⊗ d u a − 4 b d b ⊗ d b + ( d b ∘ J ) ⊗ ( d b ∘ J ) ) (40) then gives, by taking its ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 2 , 0 ) (2,0) ( 2 , 0 )
∫ B 8 ( y ) ( ∣ i ∂ ∂ ˉ log b ∣ 2 + ∣ D ′ ∂ log b + ∂ log b ⊗ ∂ log b ∣ 2 ) d V g ≤ C E . (41) \int_{B_8(y)}\left(|\mathrm{i}\partial\bar\partial\log b|^2+\left|D'\partial\log b+\partial\log b\otimes\partial\log b\right|^2\right)dV_g\le CE. \tag*{(41)} ∫ B 8 ( y ) ( ∣ i ∂ ∂ ˉ log b ∣ 2 + ∣ D ′ ∂ log b + ∂ log b ⊗ ∂ log b ∣ 2 ) d V g ≤ C E . (41) Here the tensor in square brackets in (40) is J J J b ≥ c b\ge c b ≥ c (30) , the Hessian bounds for u u u (41) , we have
∫ B 8 ( y ) ( ∣ ∂ ˉ s 0 ∣ 2 + ∣ D ′ s 0 + ∂ log b ⊗ s 0 ∣ 2 ) d V g ≤ C E . (42) \int_{B_8(y)}\left(|\bar\partial s_0|^2+\left|D's_0+\partial\log b\otimes s_0\right|^2\right)dV_g\le CE. \tag*{(42)} ∫ B 8 ( y ) ( ∣ ∂ ˉ s 0 ∣ 2 + ∣ D ′ s 0 + ∂ log b ⊗ s 0 ∣ 2 ) d V g ≤ C E . (42) The Gram determinant formula for (30) , the boundedness of its factors, and the preceding Gram estimates also give
∫ B 8 ( y ) ( b ∣ s 0 ∣ 2 − 1 ) 2 d V g ≤ C E . (43) \int_{B_8(y)}(b|s_0|^2-1)^2dV_g\le CE. \tag*{(43)} ∫ B 8 ( y ) ( b ∣ s 0 ∣ 2 − 1 ) 2 d V g ≤ C E . (43) On the Stein coordinate ball B 7 0 ( 0 ) B_7^0(0) B 7 0 ( 0 ) L 2 ∂ ˉ L^2\bar\partial L 2 ∂ ˉ [12 ] to the ordinary ( m , 1 ) (m,1) ( m , 1 ) ∂ ˉ s 0 \bar\partial s_0 ∂ ˉ s 0 e − ρ e^{-\rho} e − ρ i ∂ ∂ ˉ ρ = ω \mathrm{i}\partial\bar\partial\rho=\omega i ∂ ∂ ˉ ρ = ω ∣ ρ ∣ ≤ C 0 |\rho|\le C_0 ∣ ρ ∣ ≤ C 0
∂ ˉ ξ = ∂ ˉ s 0 , \bar\partial\xi=\bar\partial s_0, ∂ ˉ ξ = ∂ ˉ s 0 , ∫ B 7 0 ( 0 ) ∣ ξ ∣ 2 d V g ≤ e 2 C 0 ∫ B 7 0 ( 0 ) ∣ ∂ ˉ s 0 ∣ 2 d V g ≤ C E . (44) \int_{B_7^0(0)}|\xi|^2dV_g\le e^{2C_0}\int_{B_7^0(0)}|\bar\partial s_0|^2dV_g\le CE. \tag*{(44)} ∫ B 7 0 ( 0 ) ∣ ξ ∣ 2 d V g ≤ e 2 C 0 ∫ B 7 0 ( 0 ) ∣ ∂ ˉ s 0 ∣ 2 d V g ≤ C E . (44) Thus s = s 0 − ξ s=s_0-\xi s = s 0 − ξ ξ \xi ξ
On the canonical bundle, the Kähler identities give
2 ∂ ˉ ∗ ∂ ˉ = D ∗ D + S 2 ; (45) 2\bar\partial^*\bar\partial=D^*D+\frac{S}{2}; \tag*{(45)} 2 ∂ ˉ ∗ ∂ ˉ = D ∗ D + 2 S ; (45) see [12 ] . Test against ξ \xi ξ B 13 / 2 0 ( 0 ) B_{13/2}^0(0) B 13/2 0 ( 0 ) B 7 0 ( 0 ) B_7^0(0) B 7 0 ( 0 )
∫ B 6 ( y ) ( ∣ D ξ ∣ 2 + ( S + n ε ) ∣ ξ ∣ 2 ) d V g ≤ C ∫ B 7 0 ( 0 ) ( ∣ ∂ ˉ s 0 ∣ 2 + ∣ ξ ∣ 2 ) d V g ≤ C E . (46) \int_{B_6(y)}\left(|D\xi|^2+(S+n\varepsilon)|\xi|^2\right)\,dV_g
\leq C\int_{B_7^0(0)}\left(|\bar\partial s_0|^2+|\xi|^2\right)\,dV_g\leq CE. \tag*{(46)} ∫ B 6 ( y ) ( ∣ D ξ ∣ 2 + ( S + n ε ) ∣ ξ ∣ 2 ) d V g ≤ C ∫ B 7 0 ( 0 ) ( ∣ ∂ ˉ s 0 ∣ 2 + ∣ ξ ∣ 2 ) d V g ≤ C E . (46) The bound b ≥ C 0 − 1 b\geq C_0^{-1} b ≥ C 0 − 1 (42) , and ∣ d log b ∣ ≤ C |d\log b|\leq C ∣ d log b ∣ ≤ C
∫ B 6 ( y ) ( S + n ε ) ∣ s ∣ 2 + ∣ D ′ s + ∂ log b ⊗ s ∣ 2 d V g ≤ C E . (47) \int_{B_6(y)}(S+n\varepsilon)|s|^2+|D's+\partial\log b\otimes s|^2\,dV_g\leq CE. \tag*{(47)} ∫ B 6 ( y ) ( S + n ε ) ∣ s ∣ 2 + ∣ D ′ s + ∂ log b ⊗ s ∣ 2 d V g ≤ C E . (47) Moreover, ∣ ∣ s ∣ 2 − ∣ s 0 ∣ 2 ∣ ≤ 2 ∣ s 0 ∣ ∣ ξ ∣ + ∣ ξ ∣ 2 \bigl||s|^2-|s_0|^2\bigr|\leq2|s_0||\xi|+|\xi|^2 ∣ s ∣ 2 − ∣ s 0 ∣ 2 ≤ 2∣ s 0 ∣∣ ξ ∣ + ∣ ξ ∣ 2
∫ B 6 ( y ) ( S + n ε ) b ∣ ∣ s ∣ 2 − ∣ s 0 ∣ 2 ∣ d V g ≤ C ( ∫ B 6 ( y ) ( S + n ε ) ∣ s 0 ∣ 2 d V g ) 1 / 2 ( ∫ B 6 ( y ) ( S + n ε ) ∣ ξ ∣ 2 d V g ) 1 / 2 + C ∫ B 6 ( y ) ( S + n ε ) ∣ ξ ∣ 2 d V g ≤ C E . \begin{aligned}
\int_{B_6(y)}(S+n\varepsilon)b\bigl||s|^2-|s_0|^2\bigr|\,dV_g
&\leq C\left(\int_{B_6(y)}(S+n\varepsilon)|s_0|^2\,dV_g\right)^{1/2}
\left(\int_{B_6(y)}(S+n\varepsilon)|\xi|^2\,dV_g\right)^{1/2}\\
&\quad+C\int_{B_6(y)}(S+n\varepsilon)|\xi|^2\,dV_g\leq CE.
\end{aligned} ∫ B 6 ( y ) ( S + n ε ) b ∣ s ∣ 2 − ∣ s 0 ∣ 2 d V g ≤ C ( ∫ B 6 ( y ) ( S + n ε ) ∣ s 0 ∣ 2 d V g ) 1/2 ( ∫ B 6 ( y ) ( S + n ε ) ∣ ξ ∣ 2 d V g ) 1/2 + C ∫ B 6 ( y ) ( S + n ε ) ∣ ξ ∣ 2 d V g ≤ C E . Writing s = 2 − m / 2 a ( z ) d z 1 ∧ ⋯ ∧ d z m s=2^{-m/2}a(z)\,dz_1\wedge\cdots\wedge dz_m s = 2 − m /2 a ( z ) d z 1 ∧ ⋯ ∧ d z m [24 ] , we have
∣ s ∣ 2 = ∣ a ∣ 2 det ( g a b ˉ ) , ∣ s ∣ 2 d V g = ∣ a ∣ 2 d V 0 . (48) |s|^2=\frac{|a|^2}{\det(g_{a\bar b})},\qquad|s|^2dV_g=|a|^2dV_0. \tag*{(48)} ∣ s ∣ 2 = det ( g a b ˉ ) ∣ a ∣ 2 , ∣ s ∣ 2 d V g = ∣ a ∣ 2 d V 0 . (48) The bounds for s 0 s_0 s 0 ξ \xi ξ
∫ B 7 0 ( 0 ) ∣ a ∣ 2 d V 0 ≤ C , sup B 6 ( y ) ( ∣ a ∣ + ∣ d a ∣ 0 + ∣ s ∣ + ∣ ξ ∣ ) ≤ C . \int_{B_7^0(0)}|a|^2\,dV_0\leq C,\qquad\sup_{B_6(y)}\bigl(|a|+|da|_0+|s|+|\xi|\bigr)\leq C. ∫ B 7 0 ( 0 ) ∣ a ∣ 2 d V 0 ≤ C , B 6 ( y ) sup ( ∣ a ∣ + ∣ d a ∣ 0 + ∣ s ∣ + ∣ ξ ∣ ) ≤ C . It follows from (43) and s − s 0 = − ξ s-s_0=-\xi s − s 0 = − ξ
∫ B 6 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≤ C ∫ B 6 ( y ) ( ( b ∣ s 0 ∣ 2 − 1 ) 2 + ∣ ξ ∣ 2 ) d V g ≤ C E . \int_{B_6(y)}(b|s|^2-1)^2\,dV_g\leq C\int_{B_6(y)}\left((b|s_0|^2-1)^2+|\xi|^2\right)\,dV_g\leq CE. ∫ B 6 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≤ C ∫ B 6 ( y ) ( ( b ∣ s 0 ∣ 2 − 1 ) 2 + ∣ ξ ∣ 2 ) d V g ≤ C E . Choose γ > 0 \gamma>0 γ > 0 C 0 m + 1 γ 2 ≤ 1 / 4 C_0^{m+1}\gamma^2\leq1/4 C 0 m + 1 γ 2 ≤ 1/4 ∣ a ( z ∗ ) ∣ ≤ γ / 2 |a(z_*)|\leq\gamma/2 ∣ a ( z ∗ ) ∣ ≤ γ /2 z ∗ ∈ B 4 ( y ) z_*\in B_4(y) z ∗ ∈ B 4 ( y ) B ρ ∗ 0 ( z ∗ ) ⊂ B 6 ( y ) B_{\rho_*}^0(z_*)\subset B_6(y) B ρ ∗ 0 ( z ∗ ) ⊂ B 6 ( y ) ρ ∗ = ρ ∗ ( n , C 0 ) > 0 \rho_*=\rho_*(n,C_0)>0 ρ ∗ = ρ ∗ ( n , C 0 ) > 0 ∣ a ∣ ≤ γ |a|\leq\gamma ∣ a ∣ ≤ γ b ∣ s ∣ 2 ≤ 1 / 4 b|s|^2\leq1/4 b ∣ s ∣ 2 ≤ 1/4 d V g ≥ C 0 − m d V 0 dV_g\geq C_0^{-m}dV_0 d V g ≥ C 0 − m d V 0
∫ B 6 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≥ 9 16 C 0 − m Vol 0 ( B ρ ∗ 0 ) > 0. \int_{B_6(y)}(b|s|^2-1)^2\,dV_g\geq\frac{9}{16}C_0^{-m}\operatorname{Vol}_0(B_{\rho_*}^0)>0. ∫ B 6 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≥ 16 9 C 0 − m Vol 0 ( B ρ ∗ 0 ) > 0. This contradicts the preceding bound for E ≤ E 0 ( n , C 0 ) E\leq E_0(n,C_0) E ≤ E 0 ( n , C 0 ) ∣ a ∣ ≥ γ / 2 |a|\geq\gamma/2 ∣ a ∣ ≥ γ /2 B 4 ( y ) B_4(y) B 4 ( y )
Restoring the original metric multiplies ∣ s 0 ∣ |s_0| ∣ s 0 ∣ ∣ s ∣ |s| ∣ s ∣ ∣ a ∣ |a| ∣ a ∣ r − 1 r^{-1} r − 1 b b b r 2 r^2 r 2 r r r (34) –(36) and the coefficient bound (37) . □
Proof of Lemma 3.3. Rescale g , u , h g,u,h g , u , h r − 2 g , r − 1 u , r − 2 h r^{-2}g,r^{-1}u,r^{-2}h r − 2 g , r − 1 u , r − 2 h r r r ε = κ r 2 ≤ 1 \varepsilon=\kappa r^2\leq1 ε = κ r 2 ≤ 1 E E E C ( E + ε 2 ) C(E+\varepsilon^2) C ( E + ε 2 )
Apply Lemma 3.4 to obtain the nonvanishing holomorphic section s s s B 4 ( y ) B_4(y) B 4 ( y ) ω 0 ≤ C ω \omega_0\leq C\omega ω 0 ≤ C ω 2 π Ric ( ω ) + ε ω ≥ 0 2\pi\operatorname{Ric}(\omega)+\varepsilon\omega\geq0 2 π Ric ( ω ) + ε ω ≥ 0 (34) gives
∣ ∫ χ b ( ∣ s ∣ 2 − ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + ε ω ) ∧ ω 0 m − 1 ∣ ≤ C ∫ B 4 ( y ) ( S + n ε ) b ∣ ∣ s ∣ 2 − ∣ s 0 ∣ 2 ∣ d V g ≤ C E . \left|\int\chi b\left(|s|^2-|s_0|^2\right)\left(2\pi\operatorname{Ric}(\omega)+\varepsilon\omega\right)\wedge\omega_0^{m-1}\right|
\leq C\int_{B_4(y)}(S+n\varepsilon)b\left||s|^2-|s_0|^2\right|\,dV_g\leq CE. ∫ χ b ( ∣ s ∣ 2 − ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + ε ω ) ∧ ω 0 m − 1 ≤ C ∫ B 4 ( y ) ( S + n ε ) b ∣ s ∣ 2 − ∣ s 0 ∣ 2 d V g ≤ C E . Thus it remains to estimate the cutoff integral with s s s s 0 s_0 s 0
Write s = 2 − m / 2 a d z 1 ∧ ⋯ ∧ d z m s=2^{-m/2}a\,dz_1\wedge\cdots\wedge dz_m s = 2 − m /2 a d z 1 ∧ ⋯ ∧ d z m a a a (48) gives
i ∂ ∂ ˉ log ∣ s ∣ 2 = 2 π Ric ( ω ) . i\partial\bar{\partial}\log|s|^2=2\pi\operatorname{Ric}(\omega). i ∂ ∂ ˉ log ∣ s ∣ 2 = 2 π Ric ( ω ) . Consequently,
i ∂ ∂ ˉ log ( b ∣ s ∣ 2 ) = 2 π Ric ( ω ) + i ∂ ∂ ˉ log b , i\partial\bar{\partial}\log(b|s|^2)=2\pi\operatorname{Ric}(\omega)+i\partial\bar{\partial}\log b, i ∂ ∂ ˉ log ( b ∣ s ∣ 2 ) = 2 π Ric ( ω ) + i ∂ ∂ ˉ log b , i ∂ ∂ ˉ ( b ∣ s ∣ 2 ) = b ∣ s ∣ 2 i ∂ ∂ ˉ log ( b ∣ s ∣ 2 ) + i b ∣ s ∣ 2 ∂ log ( b ∣ s ∣ 2 ) ∧ ∂ ˉ log ( b ∣ s ∣ 2 ) . i\partial\bar{\partial}(b|s|^2)
=b|s|^2i\partial\bar{\partial}\log(b|s|^2)
+ib|s|^2\partial\log(b|s|^2)\wedge\bar{\partial}\log(b|s|^2). i ∂ ∂ ˉ ( b ∣ s ∣ 2 ) = b ∣ s ∣ 2 i ∂ ∂ ˉ log ( b ∣ s ∣ 2 ) + ib ∣ s ∣ 2 ∂ log ( b ∣ s ∣ 2 ) ∧ ∂ ˉ log ( b ∣ s ∣ 2 ) . Subtracting the second identity from the first and adding ε ( 1 − b ∣ s ∣ 2 ) ω \varepsilon(1-b|s|^2)\omega ε ( 1 − b ∣ s ∣ 2 ) ω
( 1 − b ∣ s ∣ 2 ) ( 2 π Ric ( ω ) + ε ω ) = ( b ∣ s ∣ 2 − 1 ) i ∂ ∂ ˉ log b + i b ∣ s ∣ 2 ∂ log ( b ∣ s ∣ 2 ) ∧ ∂ ˉ log ( b ∣ s ∣ 2 ) + ε ( 1 − b ∣ s ∣ 2 ) ω − i ∂ ∂ ˉ [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] . (49) \begin{aligned}
(1-b|s|^2)(2\pi\operatorname{Ric}(\omega)+\varepsilon\omega)
={}&(b|s|^2-1)i\partial\bar{\partial}\log b\\
&+ib|s|^2\partial\log(b|s|^2)\wedge\bar{\partial}\log(b|s|^2)\\
&+\varepsilon(1-b|s|^2)\omega\\
&-i\partial\bar{\partial}\left[b|s|^2-1-\log(b|s|^2)\right]. \tag*{(49)}
\end{aligned} ( 1 − b ∣ s ∣ 2 ) ( 2 π Ric ( ω ) + ε ω ) = ( b ∣ s ∣ 2 − 1 ) i ∂ ∂ ˉ log b + ib ∣ s ∣ 2 ∂ log ( b ∣ s ∣ 2 ) ∧ ∂ ˉ log ( b ∣ s ∣ 2 ) + ε ( 1 − b ∣ s ∣ 2 ) ω − i ∂ ∂ ˉ [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] . (49) We integrate this identity against χ ω 0 m − 1 \chi\omega_0^{m-1} χ ω 0 m − 1
For the first term, by (35) and (36) , we have
∫ B 3 ( y ) ∣ ∣ b ∣ s ∣ 2 − 1 ∣ ∣ i ∂ ∂ ˉ log b ∣ d V g ≤ ( ∫ B 4 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ) 1 / 2 ( ∫ B 4 ( y ) ∣ i ∂ ∂ ˉ log b ∣ 2 d V g ) 1 / 2 ≤ C E . \int_{B_3(y)}\left||b|s|^2-1\right|\left|i\partial\bar{\partial}\log b\right|\,dV_g
\leq
\left(\int_{B_4(y)}(b|s|^2-1)^2\,dV_g\right)^{1/2}
\left(\int_{B_4(y)}|i\partial\bar{\partial}\log b|^2\,dV_g\right)^{1/2}
\leq CE. ∫ B 3 ( y ) ∣ b ∣ s ∣ 2 − 1 i ∂ ∂ ˉ log b d V g ≤ ( ∫ B 4 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ) 1/2 ( ∫ B 4 ( y ) ∣ i ∂ ∂ ˉ log b ∣ 2 d V g ) 1/2 ≤ C E . The bound ω 0 ≤ C ω \omega_0\leq C\omega ω 0 ≤ C ω
For the second term, the Chern connection formula [12 ] gives
D ′ s + ∂ log b ⊗ s = ∂ log ( b ∣ s ∣ 2 ) ⊗ s , D's+\partial\log b\otimes s=\partial\log(b|s|^2)\otimes s, D ′ s + ∂ log b ⊗ s = ∂ log ( b ∣ s ∣ 2 ) ⊗ s , b ∣ s ∣ 2 ∣ ∂ log ( b ∣ s ∣ 2 ) ∣ 2 = b ∣ D ′ s + ∂ log b ⊗ s ∣ 2 . (50) b|s|^2|\partial\log(b|s|^2)|^2=b|D's+\partial\log b\otimes s|^2. \tag*{(50)} b ∣ s ∣ 2 ∣ ∂ log ( b ∣ s ∣ 2 ) ∣ 2 = b ∣ D ′ s + ∂ log b ⊗ s ∣ 2 . (50) Its nonnegative cutoff integral is therefore bounded by
C ∫ B 3 ( y ) b ∣ s ∣ 2 ∣ ∂ log ( b ∣ s ∣ 2 ) ∣ 2 d V g = C ∫ B 3 ( y ) b ∣ D ′ s + ∂ log b ⊗ s ∣ 2 d V g ≤ C E C\int_{B_3(y)} b|s|^2\left|\partial\log(b|s|^2)\right|^2\,dV_g
= C\int_{B_3(y)} b|D's+\partial\log b\otimes s|^2\,dV_g
\le CE C ∫ B 3 ( y ) b ∣ s ∣ 2 ∂ log ( b ∣ s ∣ 2 ) 2 d V g = C ∫ B 3 ( y ) b ∣ D ′ s + ∂ log b ⊗ s ∣ 2 d V g ≤ C E by (36) . For the third term, the volume bound and (35) imply
ε ∫ B 3 ( y ) ∣ 1 − b ∣ s ∣ 2 ∣ d V g ≤ C ε E ≤ C ( E + ε 2 ) . \varepsilon\int_{B_3(y)}\left|1-b|s|^2\right|\,dV_g
\le C\varepsilon\sqrt{E}\le C(E+\varepsilon^2). ε ∫ B 3 ( y ) 1 − b ∣ s ∣ 2 d V g ≤ C ε E ≤ C ( E + ε 2 ) . For the last term, use
0 ≤ t ( t − 1 − log t ) ≤ ( t − 1 ) 2 ( t > 0 ) , 0\le t(t-1-\log t)\le(t-1)^2\qquad(t>0), 0 ≤ t ( t − 1 − log t ) ≤ ( t − 1 ) 2 ( t > 0 ) , which follows from ( t − 1 ) 2 − t ( t − 1 − log t ) = t log t − t + 1 ≥ 0 (t-1)^2-t(t-1-\log t)=t\log t-t+1\ge0 ( t − 1 ) 2 − t ( t − 1 − log t ) = t log t − t + 1 ≥ 0 (48) , (37) , and b ≥ c b\ge c b ≥ c
[ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] d V 0 = b ∣ s ∣ 2 [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] b ∣ a ∣ 2 d V g ≤ C ( b ∣ s ∣ 2 − 1 ) 2 d V g . (51) \left[b|s|^2-1-\log(b|s|^2)\right]\,dV_0
=
\frac{b|s|^2\left[b|s|^2-1-\log(b|s|^2)\right]}{b|a|^2}\,dV_g
\le C(b|s|^2-1)^2\,dV_g. \tag*{(51)} [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] d V 0 = b ∣ a ∣ 2 b ∣ s ∣ 2 [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] d V g ≤ C ( b ∣ s ∣ 2 − 1 ) 2 d V g . (51) In particular, the first factor t = b ∣ s ∣ 2 t=b|s|^2 t = b ∣ s ∣ 2
∣ ∫ χ i ∂ ∂ ˉ [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] ∧ ω 0 m − 1 ∣ = ∣ ∫ [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] i ∂ ∂ ˉ χ ∧ ω 0 m − 1 ∣ ≤ C ∫ B 3 ( y ) [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] d V 0 ≤ C ∫ B 3 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≤ C E . \left|\int\chi\,i\partial\bar\partial\left[b|s|^2-1-\log(b|s|^2)\right]\wedge\omega_0^{m-1}\right|
=
\left|\int\left[b|s|^2-1-\log(b|s|^2)\right]i\partial\bar\partial\chi\wedge\omega_0^{m-1}\right|
\le C\int_{B_3(y)}\left[b|s|^2-1-\log(b|s|^2)\right]\,dV_0
\le C\int_{B_3(y)}(b|s|^2-1)^2\,dV_g\le CE. ∫ χ i ∂ ∂ ˉ [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] ∧ ω 0 m − 1 = ∫ [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] i ∂ ∂ ˉ χ ∧ ω 0 m − 1 ≤ C ∫ B 3 ( y ) [ b ∣ s ∣ 2 − 1 − log ( b ∣ s ∣ 2 ) ] d V 0 ≤ C ∫ B 3 ( y ) ( b ∣ s ∣ 2 − 1 ) 2 d V g ≤ C E . Here the coordinate Hessian bound for χ \chi χ C d V 0 C\,dV_0 C d V 0 b ∣ s ∣ 2 b|s|^2 b ∣ s ∣ 2
Combining these four estimates in (49) , and then using the replacement estimate at the beginning of the proof, gives
∣ ∫ χ ( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + ε ω ) ∧ ω 0 m − 1 ∣ ≤ C ( E + ε 2 ) . \left|\int\chi(1-b|s_0|^2)(2\pi\operatorname{Ric}(\omega)+\varepsilon\omega)\wedge\omega_0^{m-1}\right|\le C(E+\varepsilon^2). ∫ χ ( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + ε ω ) ∧ ω 0 m − 1 ≤ C ( E + ε 2 ) . Restoring the original scale multiplies this integral by r n − 2 r^{n-2} r n − 2 ε = κ r 2 \varepsilon=\kappa r^2 ε = κ r 2 (33) . □
Completion of the local estimate
We combine the two local estimates to prove Proposition 3.1. Positivity and the bounded Kähler potential compare the scalar integral with the coordinate cutoff integrals. The same comparison also gives the coarse bound needed when E E E
Proof of Proposition 3.1. The coordinate distortion and a first-exit argument give
B r ( y ) ⊂ B 5 r / 4 0 ( 0 ) ⊂ B 3 r / 2 0 ( 0 ) ⊂ B 2 r ( y ) . B_r(y)\subset B^0_{5r/4}(0)\subset B^0_{3r/2}(0)\subset B_{2r}(y). B r ( y ) ⊂ B 5 r /4 0 ( 0 ) ⊂ B 3 r /2 0 ( 0 ) ⊂ B 2 r ( y ) . The Ricci form is nonnegative after the shift, and
( 2 π Ric ( ω ) + κ ω ) ∧ ω m − 1 = S + n κ 2 m ω m . (2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega^{m-1}=\frac{S+n\kappa}{2m}\omega^{m}. ( 2 π Ric ( ω ) + κω ) ∧ ω m − 1 = 2 m S + nκ ω m . Apply the Chern–Levine–Nirenberg inequality [12 ] on the displayed coordinate balls. The bound ∣ ρ ∣ ≤ C 0 r 2 |\rho|\leq C_0r^2 ∣ ρ ∣ ≤ C 0 r 2 r − 2 r^{-2} r − 2
∫ B r ( y ) ( S + n κ ) d V g ≤ C ∫ B 3 r / 2 0 ( 0 ) ( 2 π Ric ( ω ) + κ ω ) ∧ ω 0 m − 1 ≤ C ∫ χ ( 2 π Ric ( ω ) + κ ω ) ∧ ω 0 m − 1 . (52) \begin{aligned}
\int_{B_r(y)}(S+n\kappa)\,dV_g\leq C\int_{B^0_{3r/2}(0)}(2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega_0^{m-1} \\
&\leq C\int\chi(2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega_0^{m-1}. \tag*{(52)}
\end{aligned} ∫ B r ( y ) ( S + nκ ) d V g ≤ C ∫ B 3 r /2 0 ( 0 ) ( 2 π Ric ( ω ) + κω ) ∧ ω 0 m − 1 ≤ C ∫ χ ( 2 π Ric ( ω ) + κω ) ∧ ω 0 m − 1 . (52) All constants below depend only on n n n C 0 C_0 C 0 E ≤ E 0 ( n , C 0 ) E\leq E_0(n,C_0) E ≤ E 0 ( n , C 0 ) Lemmas 3.3 and 3.4 apply. For the weighted term, ω 0 ≤ C ω \omega_0\leq C\omega ω 0 ≤ C ω Lemma 3.2 give
∫ χ b ∣ s 0 ∣ 2 ( 2 π Ric ( ω ) + κ ω ) ∧ ω 0 m − 1 ≤ C ∫ B 8 r ( y ) [ ∑ a ( Ric + κ g ) ( ∇ u a , ∇ u a ) + r − 2 ( Ric + κ g ) ( ∇ h , ∇ h ) ] d V g ≤ C r n − 2 E . \begin{aligned}
\int\chi b|s_0|^2(2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega_0^{m-1} \\
&\leq C\int_{B_{8r}(y)}\left[\sum_a(\operatorname{Ric}+\kappa g)(\nabla u^a,\nabla u^a)+r^{-2}(\operatorname{Ric}+\kappa g)(\nabla h,\nabla h)\right]\,dV_g\leq Cr^{n-2}E.
\end{aligned} ∫ χ b ∣ s 0 ∣ 2 ( 2 π Ric ( ω ) + κω ) ∧ ω 0 m − 1 ≤ C ∫ B 8 r ( y ) [ a ∑ ( Ric + κ g ) ( ∇ u a , ∇ u a ) + r − 2 ( Ric + κ g ) ( ∇ h , ∇ h ) ] d V g ≤ C r n − 2 E . Use Lemma 3.3 for the remaining term. Together with ∣ Ric ∣ ≤ S + 2 n κ |\operatorname{Ric}|\leq S+2n\kappa ∣ Ric ∣ ≤ S + 2 nκ (52) , and volume comparison, this yields
∫ B r ( y ) ∣ Ric ∣ d V g ≤ C ∫ χ b ∣ s 0 ∣ 2 ( 2 π Ric ( ω ) + κ ω ) ∧ ω 0 m − 1 + C ∣ ∫ χ ( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + κ ω ) ∧ ω 0 m − 1 ∣ + C κ r n ≤ C r n − 2 ( E + κ 2 r 4 ) + C κ r n ≤ C r n − 2 ( E + κ r 2 ) , (53) \begin{aligned}
\int_{B_r(y)}|\operatorname{Ric}|\,dV_g\leq C\int\chi b|s_0|^2(2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega_0^{m-1} \\
&+C\left|\int\chi(1-b|s_0|^2)(2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega_0^{m-1}\right|+C\kappa r^n \\
&\leq Cr^{n-2}(E+\kappa^2r^4)+C\kappa r^n \\
&\leq Cr^{n-2}(E+\kappa r^2), \tag*{(53)}
\end{aligned} ∫ B r ( y ) ∣ Ric ∣ d V g ≤ C ∫ χ b ∣ s 0 ∣ 2 ( 2 π Ric ( ω ) + κω ) ∧ ω 0 m − 1 + C ∫ χ ( 1 − b ∣ s 0 ∣ 2 ) ( 2 π Ric ( ω ) + κω ) ∧ ω 0 m − 1 + C κ r n ≤ C r n − 2 ( E + κ 2 r 4 ) + C κ r n ≤ C r n − 2 ( E + κ r 2 ) , (53) since κ r 2 ≤ 1 \kappa r^2\leq1 κ r 2 ≤ 1
For completeness, the chart gives a coarse bound without any smallness of E E E G = ( g a b ˉ ) G=(g_{a\bar b}) G = ( g a b ˉ ) f = log det G f=\log\det G f = log det G
∣ f ∣ ≤ det G + m log C 0 , tr G ≤ m C 0 m − 1 det G . |f|\leq\det G+m\log C_0,\qquad\operatorname{tr}G\leq mC_0^{m-1}\det G. ∣ f ∣ ≤ det G + m log C 0 , tr G ≤ m C 0 m − 1 det G . Since d V g = det G d V 0 dV_g=\det G\,dV_0 d V g = det G d V 0
∫ B 3 r ( y ) ( ∣ f ∣ + tr G ) d V 0 ≤ C r n . \int_{B_{3r}(y)}(|f|+\operatorname{tr}G)\,dV_0\leq Cr^n. ∫ B 3 r ( y ) ( ∣ f ∣ + tr G ) d V 0 ≤ C r n . Using 2 π Ric ( ω ) = − i ∂ ∂ ˉ f 2\pi\operatorname{Ric}(\omega)=-i\partial\bar\partial f 2 π Ric ( ω ) = − i ∂ ∂ ˉ f
∫ χ ( 2 π Ric ( ω ) + κ ω ) ∧ ω 0 m − 1 = − ∫ f i ∂ ∂ ˉ χ ∧ ω 0 m − 1 + κ ∫ χ ω ∧ ω 0 m − 1 ≤ C r − 2 ∫ B 3 r ( y ) ∣ f ∣ d V 0 + C κ ∫ B 3 r ( y ) tr G d V 0 ≤ C r n − 2 . \begin{aligned}
\int\chi(2\pi\operatorname{Ric}(\omega)+\kappa\omega)\wedge\omega_0^{m-1}
=-\int f\,i\partial\bar\partial\chi\wedge\omega_0^{m-1}+\kappa\int\chi\omega\wedge\omega_0^{m-1} \\
&\leq Cr^{-2}\int_{B_{3r}(y)}|f|\,dV_0+C\kappa\int_{B_{3r}(y)}\operatorname{tr}G\,dV_0 \\
&\leq Cr^{n-2}.
\end{aligned} ∫ χ ( 2 π Ric ( ω ) + κω ) ∧ ω 0 m − 1 = − ∫ f i ∂ ∂ ˉ χ ∧ ω 0 m − 1 + κ ∫ χ ω ∧ ω 0 m − 1 ≤ C r − 2 ∫ B 3 r ( y ) ∣ f ∣ d V 0 + C κ ∫ B 3 r ( y ) tr G d V 0 ≤ C r n − 2 . Thus (52) gives ∫ B r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 \int_{B_r(y)}|\operatorname{Ric}|\,dV_g\leq Cr^{n-2} ∫ B r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 E > E 0 E>E_0 E > E 0 C E 0 − 1 r n − 2 E CE_0^{-1}r^{n-2}E C E 0 − 1 r n − 2 E (29) in this case as well. □
Proof of the main theorem
We now complete the proof of Theorem 1.1, following the neck summation argument of Jiang–Naber [18 ] . The new input is the local Ricci L 1 L^1 L 1 § 3 . Combined with Proposition 2.17 , it gives the required entropy estimate on regular balls, which can then be summed by the neck structure and entropy monotonicity.
Fix η = η ( n , v ) > 0 \eta=\eta(n,v)>0 η = η ( n , v ) > 0 a 0 a_0 a 0 L L L δ 0 \delta_0 δ 0 n = 2 m ≥ 4 n=2m\geq4 n = 2 m ≥ 4
The next proposition gives the L 1 L^1 L 1 Theorem 2.4 and telescoping of the entropy drops, with Theorem 2.14 handling the remaining almost Euclidean balls.
Proposition 4.1. For n = 2 m ≥ 4 n=2m\geq4 n = 2 m ≥ 4 v > 0 v>0 v > 0 η = η ( n , v ) > 0 \eta=\eta(n,v)>0 η = η ( n , v ) > 0 δ = δ ( n , v ) > 0 \delta=\delta(n,v)>0 δ = δ ( n , v ) > 0 N = B 2 R ( q ) ∖ B ‾ r x ( C ) \mathcal{N}=B_{2R}(q)\setminus\overline{B}_{r_x}(C) N = B 2 R ( q ) ∖ B r x ( C ) ( n − 2 , δ 0 , η ) (n-2,\delta_0,\eta) ( n − 2 , δ 0 , η ) ( M n , g , J ) (M^n,g,J) ( M n , g , J ) 0 < δ 0 ≤ δ 0<\delta_0\leq\delta 0 < δ 0 ≤ δ δ 1 > 0 \delta_1>0 δ 1 > 0
Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q ) ) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) , δ 1 R ≤ 4 δ 0 ≤ 1 100 . \operatorname{Ric}_g\geq-(n-1)\delta_1^2g,\qquad\operatorname{Vol}(B_\rho(q))\geq v\rho^n\quad(0<\rho\leq\delta_1^{-1}),\qquad\delta_1R\leq4\delta_0\leq\frac{1}{100}. Ric g ≥ − ( n − 1 ) δ 1 2 g , Vol ( B ρ ( q )) ≥ v ρ n ( 0 < ρ ≤ δ 1 − 1 ) , δ 1 R ≤ 4 δ 0 ≤ 100 1 . Then
∫ N ∩ B R ( q ) ∣ Ric ∣ d V g ≤ C ( n , v ) R n − 2 . (54) \int_{\mathcal{N}\cap B_R(q)}|\operatorname{Ric}|\,dV_g\leq C(n,v)R^{n-2}. \tag*{(54)} ∫ N ∩ B R ( q ) ∣ Ric ∣ d V g ≤ C ( n , v ) R n − 2 . (54) Proof. Fix L = L ( n , v , η ) ≥ L 0 ( n , v , η ) L=L(n,v,\eta)\geq L_0(n,v,\eta) L = L ( n , v , η ) ≥ L 0 ( n , v , η ) Proposition 2.17 , and choose a 0 = a 0 ( n , v , η ) a_0=a_0(n,v,\eta) a 0 = a 0 ( n , v , η ) Corollary 2.15 . Put κ = ( n − 1 ) δ 1 2 \kappa=(n-1)\delta_1^2 κ = ( n − 1 ) δ 1 2 μ = μ N \mu=\mu_{\mathcal{N}} μ = μ N q q q
We first obtain the estimate on each regular ball. For y ∈ N ∩ B R ( q ) y\in\mathcal{N}\cap B_R(q) y ∈ N ∩ B R ( q ) z ∈ C z\in C z ∈ C
r = d ( y , C ) = d ( y , z ) > 0 , r ≤ R 10 6 L , s = L r , (55) r=d(y,C)=d(y,z)>0,\qquad r\leq\frac{R}{10^6L},\qquad s=Lr, \tag*{(55)} r = d ( y , C ) = d ( y , z ) > 0 , r ≤ 1 0 6 L R , s = L r , (55) and suppose that B a 0 r ( y ) ⊂ N B_{a_0r}(y)\subset\mathcal{N} B a 0 r ( y ) ⊂ N P P P (16) . Choose a sufficiently small c = c ( n , v , η , L ) > 0 c=c(n,v,\eta,L)>0 c = c ( n , v , η , L ) > 0 δ ( n , v ) \delta(n,v) δ ( n , v ) Proposition 2.17 applies whenever P ≤ c P\leq c P ≤ c κ r 2 ≤ 16 ( n − 1 ) δ 0 2 \kappa r^2\leq16(n-1)\delta_0^2 κ r 2 ≤ 16 ( n − 1 ) δ 0 2
For P ≤ c P\leq c P ≤ c Proposition 2.17 to B 8 a 0 r ( y ) B_{8a_0r}(y) B 8 a 0 r ( y ) Corollary 2.15 , written in the original metric, and the bounds for these functions give the local setting of § 3 with radius a 0 r a_0r a 0 r C 0 = C 0 ( n , v , η , L ) C_0=C_0(n,v,\eta,L) C 0 = C 0 ( n , v , η , L ) 8 a 0 < 1 / 8 8a_0<1/8 8 a 0 < 1/8 b b b (28) , evaluated at radius a 0 r a_0r a 0 r a 0 − n − 4 a_0^{-n-4} a 0 − n − 4 (27) . By Proposition 3.1 , we have
∫ B a 0 r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ( P + κ r 2 ) . \int_{B_{a_0r}(y)}|\operatorname{Ric}|\,dV_g\leq Cr^{n-2}(P+\kappa r^2). ∫ B a 0 r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ( P + κ r 2 ) . If P > c P>c P > c Theorem 2.14 on the same regular ball gives
∫ B a 0 r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ≤ C r n − 2 P . \int_{B_{a_0r}(y)}|\operatorname{Ric}|\,dV_g\leq Cr^{n-2}\leq Cr^{n-2}P. ∫ B a 0 r ( y ) ∣ Ric ∣ d V g ≤ C r n − 2 ≤ C r n − 2 P . The chart accuracy gives the required Euclidean approximation; volume comparison and a further decrease of δ \delta δ r n − 2 P = L 2 − n s n − 2 P r^{n-2}P=L^{2-n}s^{n-2}P r n − 2 P = L 2 − n s n − 2 P
∫ B a 0 r ( y ) ∣ Ric ∣ d V g ≤ A ∫ C ∩ B s ( z ) ( W s 2 / 2 δ 1 ( x ) − W 2 s 2 δ 1 ( x ) ) d μ ( x ) + A δ 1 2 r n , (56) \int_{B_{a_0r}(y)}|\operatorname{Ric}|\,dV_g\leq A\int_{C\cap B_s(z)}\left(\mathcal{W}_{s^2/2}^{\delta_1}(x)-\mathcal{W}_{2s^2}^{\delta_1}(x)\right)\,d\mu(x)+A\delta_1^2r^n, \tag*{(56)} ∫ B a 0 r ( y ) ∣ Ric ∣ d V g ≤ A ∫ C ∩ B s ( z ) ( W s 2 /2 δ 1 ( x ) − W 2 s 2 δ 1 ( x ) ) d μ ( x ) + A δ 1 2 r n , (56) where A = A ( n , v , η , L ) A=A(n,v,\eta,L) A = A ( n , v , η , L ) η = η ( n , v ) \eta=\eta(n,v) η = η ( n , v ) L = L ( n , v , η ) L=L(n,v,\eta) L = L ( n , v , η ) n , v n,v n , v
Put θ = ( 10 6 L ) − 1 \theta=(10^{6}L)^{-1} θ = ( 1 0 6 L ) − 1 B a 0 r i ( y i ) B_{a_0r_i}(y_i) B a 0 r i ( y i ) B a 0 r i / 5 ( y i ) B_{a_0r_i/5}(y_i) B a 0 r i /5 ( y i ) d ( y , C ) ≤ θ R d(y,C)\leq\theta R d ( y , C ) ≤ θ R d ( y , x ) > 100 r x d(y,x)>100r_x d ( y , x ) > 100 r x x ∈ C x\in C x ∈ C r i = d ( y i , C ) = d ( y i , z i ) r_i=d(y_i,C)=d(y_i,z_i) r i = d ( y i , C ) = d ( y i , z i ) s i = L r i s_i=Lr_i s i = L r i N \mathcal{N} N ∑ i r i n ≤ C R n \sum_i r_i^n\leq CR^n ∑ i r i n ≤ C R n C ∩ B s i ( z i ) C\cap B_{s_i}(z_i) C ∩ B s i ( z i ) C R n − 2 CR^{n-2} C R n − 2 Theorem 2.4 ; all its centers lie in B 3 R / 2 ( q ) B_{3R/2}(q) B 3 R /2 ( q ) r x ≤ R / 16 r_x\leq R/16 r x ≤ R /16
For ρ j = 2 − j θ R \rho_j=2^{-j}\theta R ρ j = 2 − j θ R T j = ( L ρ j ) 2 T_j=(L\rho_j)^2 T j = ( L ρ j ) 2 ρ j / 2 < r i ≤ ρ j \rho_j/2<r_i\leq\rho_j ρ j /2 < r i ≤ ρ j
0 ≤ W s i 2 / 2 δ 1 ( x ) − W 2 s i 2 δ 1 ( x ) ≤ W T j / 8 δ 1 ( x ) − W 2 T j δ 1 ( x ) . 0\leq W^{\delta_1}_{s_i^2/2}(x)-W^{\delta_1}_{2s_i^2}(x)\leq W^{\delta_1}_{T_j/8}(x)-W^{\delta_1}_{2T_j}(x). 0 ≤ W s i 2 /2 δ 1 ( x ) − W 2 s i 2 δ 1 ( x ) ≤ W T j /8 δ 1 ( x ) − W 2 T j δ 1 ( x ) . Since 2 T j + 3 < T j / 8 2T_{j+3}<T_j/8 2 T j + 3 < T j /8 j j j (56) , we have
∑ i ∫ B a 0 r i ( y i ) ∣ Ric ∣ d V g ≤ A ∑ i ∫ C ∩ B s i ( z i ) ( W s i 2 / 2 δ 1 ( x ) − W 2 s i 2 δ 1 ( x ) ) d μ ( x ) + C A δ 1 2 R n ≤ C R n − 2 . (57) \sum_i\int_{B_{a_0r_i}(y_i)}|\operatorname{Ric}|\,dV_g
\leq A\sum_i\int_{C\cap B_{s_i}(z_i)}
\left(W^{\delta_1}_{s_i^2/2}(x)-W^{\delta_1}_{2s_i^2}(x)\right)\,d\mu(x)+CA\delta_1^2R^n
\leq CR^{n-2}. \tag*{(57)} i ∑ ∫ B a 0 r i ( y i ) ∣ Ric ∣ d V g ≤ A i ∑ ∫ C ∩ B s i ( z i ) ( W s i 2 /2 δ 1 ( x ) − W 2 s i 2 δ 1 ( x ) ) d μ ( x ) + C A δ 1 2 R n ≤ C R n − 2 . (57) The remaining points have d ( y , C ) > θ R d(y,C)>\theta R d ( y , C ) > θ R r x / 2 ≤ d ( y , C ) ≤ 100 r x r_x/2\leq d(y,C)\leq100r_x r x /2 ≤ d ( y , C ) ≤ 100 r x Corollary 2.15 covers them by almost Euclidean balls at fixed multiples of θ R \theta R θ R r x r_x r x ( n − 2 ) (n-2) ( n − 2 ) C R n − 2 CR^{n-2} C R n − 2 Theorem 2.14 completes (54) . □
The bounds for necks and almost Euclidean balls now have the same r n − 2 r^{n-2} r n − 2 Theorem 2.5 completes the proof.
Proof of Theorem 1.1 . By choosing η = η ( n , v ) > 0 \eta=\eta(n,v)>0 η = η ( n , v ) > 0 ( n − 1 , 2 η ) (n-1,2\eta) ( n − 1 , 2 η ) Theorem 2.14 applies to these balls. Proposition 4.1 bounds each neck. By Theorem 2.5 , the estimate (7) gives
∫ B 1 ( p ) ∣ Ric ∣ d V g ≤ ∑ a ∫ N a ∩ B r a ( x a ) ∣ Ric ∣ d V g + ∑ b ∫ B r b ( x b ) ∣ Ric ∣ d V g ≤ C ( ∑ a r a n − 2 + ∑ b r b n − 2 ) ≤ C ( n , v ) . (58) \int_{B_1(p)}|\operatorname{Ric}|\,dV_g
\leq\sum_a\int_{\mathcal{N}_a\cap B_{r_a}(x_a)}|\operatorname{Ric}|\,dV_g+\sum_b\int_{B_{r_b}(x_b)}|\operatorname{Ric}|\,dV_g
\leq C\left(\sum_a r_a^{n-2}+\sum_b r_b^{n-2}\right)\leq C(n,v). \tag*{(58)} ∫ B 1 ( p ) ∣ Ric ∣ d V g ≤ a ∑ ∫ N a ∩ B r a ( x a ) ∣ Ric ∣ d V g + b ∑ ∫ B r b ( x b ) ∣ Ric ∣ d V g ≤ C ( a ∑ r a n − 2 + b ∑ r b n − 2 ) ≤ C ( n , v ) . (58) Acknowledgments
The authors are deeply grateful to Aaron Naber for his valuable suggestions and many discussions on this problem over the years. His ideas and insights have been an important source of inspiration for this work and have helped shape our approach to the problem. We also thank him for his encouragement and generosity in sharing his mathematical ideas.
The authors acknowledge the substantial assistance of OpenAI’s ChatGPT in developing the local linear estimate in Proposition 3.1 . In earlier attempts, the authors had obtained only a square-root estimate in the entropy error. AI-assisted exploration led to the decomposition of the scalar-curvature integral described in the “Idea of the proof” of Proposition 3.1 . This decomposition was a key step in obtaining the linear bound required for the neck summation. The authors subsequently checked and completed the argument and reorganized its presentation. They take full responsibility for the mathematical content of the paper.
This work was supported by the National Key Research and Development Program of China (No. 2022YFA1005500), the National Natural Science Foundation of China (Grant Nos. 12125105 and 12071140), the New Cornerstone Science Foundation through the New Cornerstone Investigator Program, the Xplorer Prize, and the Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014).
References
School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China Email address: wsjiang@zju.edu.cn
School of Mathematical Sciences, East China Normal University, Shanghai 200241, China Email address: gliu@math.ecnu.edu.cn
