A lower bound for the Kähler-Einstein distance from the Diederich-Fornæss index
aa r X i v : . [ m a t h . C V ] A p r A LOWER BOUND FOR THE K ¨AHLER-EINSTEIN DISTANCEFROM THE DIEDERICH-FORNÆSS INDEX
ANDREW ZIMMER
Abstract.
In this note we establish a lower bound for the distance inducedby the K¨ahler-Einstein metric on pseudoconvex domains with positive hyper-convexity index (e.g. positive Diederich-Fornæss index). A key step is provingan analog of the Hopf lemma for Riemannian manifolds with Ricci curvaturebounded from below. Introduction
Every bounded pseudoconvex domain Ω ⊂ C d has a unique complete K¨ahler-Einstein metric, denoted by g KE , with Ricci curvature − (2 d − C boundary and by Mok andYau [MY83] in general.Let d KE be the distance induced by g KE . Since g KE is complete, if we fix z ∈ Ω,then lim z → ∂ Ω d KE ( z, z ) = ∞ . (1)In this note we consider quantitative versions of Equation (1). In particular, itis natural to ask for lower bounds on d KE ( z, z ) in terms of the distance to theboundary function δ Ω ( z ) = min {k w − z k : w ∈ ∂ Ω } . Mok and Yau proved for every z ∈ Ω there exists C , C ∈ R such that d KE ( z, z ) ≥ C + C log log 1 δ Ω ( z )for all z ∈ Ω, see [MY83, pg. 47]. Further, by considering the case of a punctureddisk, this lower bound is the best possible for general pseudoconvex domains.However, for certain classes of bounded pseudoconvex domains, there are muchbetter lower bounds. For instance, if Ω is convex, then for any z ∈ Ω there exists C , C > d KE ( z, z ) ≥ C + C log 1 δ Ω ( z )(2)for all z ∈ Ω, see [Fra91]. In this note, we show that Estimate (2) holds for a largeclass of domains - those with positive hyperconvexity index.
Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA
E-mail address : [email protected] . Date : April 15, 2020.
First we recall the well studied Diederich-Fornæss index. Suppose Ω ⊂ C d isa bounded pseudoconvex domain. A number τ ∈ (0 ,
1) is a called an
Diederich-Fornæss exponent of
Ω if there exist a continuous plurisubharmonic function ψ :Ω → ( −∞ ,
0) and a constant
C >
C δ Ω ( z ) τ ≤ − ψ ( z ) ≤ Cδ Ω ( z ) τ for all z ∈ Ω. Then the
Diederich-Fornæss index of
Ω is defined to be η (Ω) := sup { τ : τ is a Diederich-Fornæss exponent of Ω } . It is known that η (Ω) > η (Ω) > ∂ Ω is C . Later, Harrington [Har08] generalized thisresult and proved that η (Ω) > ∂ Ω is Lipschitz.The hyperconvexity index, introduced by Chen [Che17], is a similar quantityassociated to a bounded pseudoconvex domain Ω ⊂ C d . In particular, a number τ ∈ (0 ,
1) is a called an hyperconvexity exponent of
Ω if there exist a continuousplurisubharmonic function ψ : Ω → ( −∞ ,
0) and a constant
C > − ψ ( z ) ≤ Cδ Ω ( z ) τ for all z ∈ Ω. Then the hyperconvex index of
Ω is defined to be α (Ω) := sup { τ : τ is a hyperconvexity exponent of Ω } . By definition α (Ω) ≥ η (Ω). Further, it is sometimes easier to verify that thehyperconvexity index is positive (see [Che17, Appendix]).For domains with positive hyperconvexity index we will establish the followinglower bound for d KE . Theorem 1.1.
Suppose Ω ⊂ C d is a bounded pseudoconvex domain with α (Ω) > .If z ∈ Ω and ǫ > , then there exists some C = C ( z , ǫ ) ≤ such that d KE ( z, z ) ≥ C + (cid:18) α (Ω)2 d − − ǫ (cid:19) log 1 δ Ω ( z ) for all z ∈ Ω . In this note we have normalized the K¨ahler-Einstein metric to have Ricci curva-ture equal to − (2 d − − (2 d − λ we would obtain the lower bound C + 1 √ λ (cid:18) α (Ω)2 d − − ǫ (cid:19) log 1 δ Ω ( z ) . In fact, we will show that Estimate (2) holds for any complete K¨ahler metricwith Ricci curvature bounded from below.
Theorem 1.2.
Suppose Ω ⊂ C d is a bounded pseudoconvex domain with α (Ω) > , g is a complete K¨ahler metric on Ω with Ric g ≥ − (2 d − , and d g is the distanceassociated to g . If z ∈ Ω and ǫ > , then there exists some C = C ( z , ǫ ) ≤ suchthat d g ( z , z ) ≥ C + (cid:18) α (Ω)2 d − − ǫ (cid:19) log 1 δ Ω ( z ) for all z ∈ Ω . OWER BOUND FOR DISTANCES 3
Lower bounds on the Bergman metric.
It is conjectured that the Bergmandistance on a bounded pseudoconvex domain with C boundary also satisfies Es-timate (2). In this direction, the best general result is due B locki [B lo05] whoextended work of Diederich-Ohsawa [DO95] and established a lower bound of theform C + C /δ Ω ( z )) log 1 δ Ω ( z )for the Bergman distance on a bounded pseudoconvex domain with C boundary.Notice that Theorem 1.2 implies the conjectured lower bound for the Bergmandistance under the additional assumption that the Ricci curvature of the Bergmanmetric is bounded from below. Acknowledgements.
I would to thank Yuan Yuan and Liyou Zhang for bring-ing the hyperconvexity index to my attention. This material is based upon worksupported by the National Science Foundation under grant DMS-1904099.2.
A Hopf Lemma for Riemannian manifolds
The standard proof of the Hopf lemma implies the following estimate:
Proposition 2.1 (Hopf Lemma) . If D ⊂ R d is a bounded domain with C boundaryand ϕ : D → ( −∞ , is subharmonic, then there exists C > such that ϕ ( x ) ≤ − Cδ D ( x ) for all x ∈ D . We will prove a variant of (this version of) the Hopf Lemma for Riemannianmanifolds with Ricci curvature bounded below.Given a complete Riemannian manifold (
X, g ), let d g denote the distance inducedby g , let ∇ g denote the gradient, and let ∆ g denote the Laplace-Beltrami operatoron X . A function ϕ : X → R is subharmonic if ∆ g ϕ ≥ Proposition 2.2.
Suppose that ( X, g ) is a complete Riemannian manifold with Ric( g ) ≥ − (2 d − . If x ∈ X , ǫ > , and ϕ : X → ( −∞ , is subharmonic, thenthere exists C > such that ϕ ( x ) ≤ − C exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) for all x ∈ X . We require one lemma. Given a complete Riemannian manifold (
X, g ), x ∈ X ,and r > B g ( x, r ) = { y ∈ X : d g ( x, y ) < r } . Lemma 2.3.
Suppose that ( X, g ) is a complete Riemannian manifold with Ric( g ) ≥− (2 d − . Then for every x ∈ X and ǫ > , there exists r > such that thefunction Φ( x ) = exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) is subharmonic on X \ B g ( x , r ) . LOWER BOUND FOR DISTANCES
When the function x → d g ( x, x ) is smooth on X \ { x } , the lemma is animmediate consequence of the Laplacian comparison theorem. We prove the generalcase by simply modifying the proof of the Laplacian comparison theorem givenin [Pet16]. Proof.
Let r ( x ) = d g ( x, x ). We will show that∆ g Φ( x ) ≥ Φ( x ) (cid:0) (2 d − ǫ ) − (2 d − d − ǫ ) coth r ( x ) (cid:1) in the sense of distributions on X \ { x } , which implies the lemma.Fix q ∈ X and let σ : [0 , T ] → X be a unit speed geodesic joining x to q .Then for δ ∈ (0 , T ) consider the function r q,δ ( x ) = d g ( x, σ ( δ )) + δ . By the proofof [Pet16, Lemma 7.1.9], q is not in the cut locus of σ ( δ ). In particular, there existsa neighborhood O q of q such that r q,δ is C ∞ and k∇ g r q,δ k ≡ O q , see [Sak96, Proposition III.4.8]. Further, by the Laplacian comparisontheorem ∆ g r q,δ ( x ) ≤ (2 d −
1) coth ( r q,δ ( x ) − δ )on O q , see [Pet16, Lemma 7.1.9]. Next consider the function Φ q,δ : O q → [0 , ∞ )defined by Φ q,δ ( x ) = exp (cid:16) − (2 d − ǫ ) r q,δ ( x ) (cid:17) . Then ∆ g Φ q,δ ( x ) = Φ q,δ ( x ) (cid:16) (2 d − ǫ ) k∇ g r q,δ k − (2 d − ǫ )∆ g r q,δ ( x ) (cid:17) ≥ Φ q,δ ( x ) (cid:0) (2 d − ǫ ) − (2 d − ǫ )(2 d −
1) coth ( r q,δ ( x ) − δ ) (cid:1) . (3)Fix a partition of unit 1 = P ∞ j =1 χ j subordinate to the open cover X = ∪ q ∈ X O q .For each j ∈ N , fix q j ∈ X such that supp( χ j ) ⊂ O q j .Now suppose that ψ : X \ { x } → [0 , ∞ ) is a compactly supported smoothfunction. Then by the dominated convergence theorem (notice that the sum isfinite) Z X Φ( x )∆ g ψ ( x ) dx = lim δ → + ∞ X j =1 Z O qj Φ q j ,δ ( x )∆ g ( χ j ( x ) ψ ( x )) dx. By integration by parts and Equation (3) Z O qj Φ q j ,δ ( x )∆ g ( χ j ( x ) ψ ( x )) dx = Z O qj χ j ( x ) ψ ( x )∆ g Φ q j ,δ ( x ) dx ≥ Z O qj χ j ( x ) ψ ( x )Φ q,δ ( x ) (cid:0) (2 d − ǫ ) − (2 d − ǫ )(2 d −
1) coth ( r q,δ ( x ) − δ ) (cid:1) dx. So by applying the dominated convergence theorem again Z X Φ( x )∆ g ψ ( x ) dx ≥ Z X Φ( x ) (cid:0) (2 d − ǫ ) − (2 d − d − ǫ ) coth r ( x ) (cid:1) ψ ( x ) dx. Hence ∆ g Φ( x ) ≥ Φ( x ) (cid:0) (2 d − ǫ ) − (2 d − d − ǫ ) coth r ( x ) (cid:1) in the sense of distributions on X \ { x } . (cid:3) OWER BOUND FOR DISTANCES 5
Proof of Proposition 2.2.
Fix r > x → exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) is subharmonic on X \ B g ( x , r ). Since ϕ <
0, there exists
C > ϕ ( x ) ≤ − C exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) for all x ∈ B g ( x , r ). Then consider f ( x ) = ϕ ( x ) + C exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) . Then f is subharmonic on X \ B g ( x , r ). Fix R > r and let A R = B g ( x , R ) \ B g ( x , r )Then f ( x ) ≤ ∂B g ( x , r ) and f ( x ) ≤ C exp (cid:16) − (2 d − ǫ ) R (cid:17) on ∂B g ( x , R ). So by the maximum principle f ( x ) ≤ C exp (cid:16) − (2 d − ǫ ) R (cid:17) on A R . Then sending R → f ( x ) ≤ X \ B g ( x , r ). So ϕ ( x ) ≤ − C exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) for all x ∈ X . (cid:3) Proof of Theorem 1.2
Suppose Ω ⊂ C d is a bounded pseudoconvex domain with α (Ω) > g is acomplete K¨ahler metric on Ω with Ric g ≥ − (2 d − z ∈ Ω, and ǫ > ǫ > τ ∈ (0 ,
1) such that τ d − ǫ ≥ α (Ω)2 d − − ǫ. Then there exists a continuous plurisubharmonic function ψ : Ω → ( −∞ ,
0) and a > − ψ ( z ) ≤ aδ Ω ( z ) τ for all z ∈ Ω.Since ψ is plurisubharmonic and g is K¨ahler, ψ is subharmonic on (Ω , g ). So byProposition 2.2 there exists C > ψ ( z ) ≤ − C exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) for all z ∈ Ω. Then − aδ Ω ( z ) τ ≤ − C exp (cid:16) − (2 d − ǫ ) d g ( x, x ) (cid:17) LOWER BOUND FOR DISTANCES and so there exists C ∈ R such that C + (cid:18) τ d − ǫ (cid:19) log 1 δ Ω ( z ) ≤ d g ( z, z )for all z ∈ Ω. Since the set { z ∈ Ω : δ Ω ( z ) ≥ } is compact and τ d − ǫ ≥ α (Ω)2 d − − ǫ, there exists C ∈ R such that C + (cid:16) α (Ω)2 d − − ǫ (cid:17) log 1 δ Ω ( z ) ≤ d g ( z, z )for all z ∈ Ω. References [B lo05] Zbigniew B locki. The Bergman metric and the pluricomplex Green function.
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