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A lower bound for the K\"ahler-Einstein distance from the Diederich-Forn{\ae}ss index

In this note we establish a lower bound for the distance induced by the K\"ahler-Einstein metric on pseudoconvex domains with positive hyperconvexity index (e.g. positive Diederich-Fornaess index). A key step is proving an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.


Introduction
Every bounded pseudoconvex domain Ω ⊂ C d has a unique complete Kähler-Einstein metric, denoted by g KE , with Ricci curvature −(2d − 1). This was constructed by Cheng and Yau [CY80] when Ω has C 2 boundary and by Mok and Yau [MY83] in general.
Let d KE be the distance induced by g KE . Since g KE is complete, if we fix z 0 ∈ Ω, then lim z→∂Ω d KE (z, z 0 ) = ∞. (1) In this note we consider quantitative versions of Equation (1). In particular, it is natural to ask for lower bounds on d KE (z, z 0 ) in terms of the distance to the boundary function δ Ω (z) = min{ w − z : w ∈ ∂Ω}.
Mok and Yau proved for every z 0 ∈ Ω there exists C 1 , C 2 ∈ R such that d KE (z, z 0 ) ≥ C 1 + C 2 log log 1 δ Ω (z) for all z ∈ Ω, see [MY83,pg. 47]. Further, by considering the case of a punctured disk, this lower bound is the best possible for general pseudoconvex domains.
However, for certain classes of bounded pseudoconvex domains, there are much better lower bounds. For instance, if Ω is convex, then for any z 0 ∈ Ω there exists C 1 , C 2 > 0 such that for all z ∈ Ω, see [Fra91]. In this note, we show that Estimate (2) holds for a large class of domains -those with positive Diederich-Fornaess index.
Theorem 1.1. Suppose Ω ⊂ C d is a bounded pseudoconvex domain with η(Ω) > 0. If z 0 ∈ Ω and ǫ > 0, then there exists some C = C(z 0 , ǫ) ≤ 0 such that In this note we have normalized the Kähler-Einstein metric to have Ricci curvature equal to −(2d − 1). If we instead normalized so that the Ricci curvature equals −(2d − 1)λ we would obtain the lower bound .
In fact, we will show that Estimate (2) holds for any complete Kähler metric with Ricci curvature bounded from below.
Theorem 1.2. Suppose Ω ⊂ C d is a bounded pseudoconvex domain with η(Ω) > 0, g is a complete Kähler metric on Ω with Ric g ≥ −(2d − 1), and d g is the distance associated to g. If z 0 ∈ Ω and ǫ > 0, then there exists some C = C(z 0 , ǫ) ≤ 0 such that for all z ∈ Ω.
1.1. Lower bounds on the Bergman metric. It is conjectured that the Bergman distance on a bounded pseudoconvex domain with C 2 boundary also satisfies Estimate (2). In this direction, the best general result is due B locki [B lo05] who extended work of Diederich-Ohsawa [DO95] and established a lower bound of the form for the Bergman distance on a bounded pseudoconvex domain with C 2 boundary.
Notice that Theorem 1.2 implies the conjectured lower bound for the Bergman distance under the additional assumption that the Ricci curvature of the Bergman metric is bounded from below.
Acknowledgements. This material is based upon work supported by the National Science Foundation under grant DMS-1904099.

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 2 boundary and ϕ : D → (−∞, 0) is subharmonic, then there exists C > 0 such that for all x ∈ D.
We will prove a variant of this version of the Hopf Lemma for Riemannian manifolds with Ricci curvature bounded below.
Given a complete Riemannian manifold (X, g), let d g denote the distance induced by g, let ∇ g denote the gradient, and let ∆ g denote the Laplace-Beltrami operator on X. A function ϕ : X → R is subharmonic if ∆ g ϕ ≥ 0 in the sense of distributions.
We require one lemma. Given a complete Riemannian manifold (X, g), x ∈ X, and r > 0 define 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) ≥ −(2d − 1). Then for every x 0 ∈ X and ǫ > 0, there exists r 0 > 0 such that the function When the function x → d g (x, x 0 ) is smooth on X \ {x 0 }, the lemma is an immediate consequence of the Laplacian comparison theorem. We prove the general case by simply modifying the proof of the Laplacian comparison theorem given in [Pet16].
in the sense of distributions on X \ {x 0 }, which implies the lemma.
Fix q ∈ X and let σ : [0, T ] → X be a unit speed geodesic joining x 0 to q. Then for δ ∈ (0, T ) consider the function r q,δ (x) = d g (x, σ(δ)) + δ. By the proof of [Pet16, Lemma 7.1.9], q is not in the cut locus of σ(δ). In particular, there exists a neighborhood O q of q such that r q,δ is C ∞ and ∇ g r q,δ ≡ 1 on O q , see [Sak96,Proposition III.4.8]. Further, by the Laplacian comparison theorem on O q , see [Pet16, Lemma 7.1.9]. Next consider the function Φ q,δ : is a compactly supported smooth function. Then by the dominated convergence theorem (notice that the sum is finite)

By integration by parts and Equation (3)
So by applying the dominated convergence theorem again in the sense of distributions on X \ {x 0 }.
Proof of Proposition 2.2. Fix r 0 > 0 such that is subharmonic on X \ B g (x 0 , r 0 ). Since ϕ < 0, there exists C > 0 such that for all x ∈ B g (x 0 , r 0 ). Then consider Then f is subharmonic on X \ B g (x 0 , r). Fix R > r 0 and let for all x ∈ X.