The Stochastic Schwarz lemma on K\"ahler Manifolds by Couplings and Its Applications

TL;DR

This paper introduces a stochastic approach to the Schwarz lemma on K"ahler manifolds using Markovian couplings, leading to new comparison results, gradient estimates, and applications to quaternionic K"ahler manifolds.

Contribution

It develops a probabilistic framework for the Schwarz lemma on K"ahler manifolds, enabling new curvature comparison results and gradient estimates not accessible by classical methods.

Findings

Established a stochastic formula for Carathéodory distance.

Proved comparison between Carathéodory distance and K"ahler metric with negative curvature.

Derived improved gradient estimates for harmonic functions.

Abstract

We first provide a stochastic formula for the Carath\'eodory distance in terms of general Markovian couplings and prove a comparison result between the Carath\'eodory distance and the complete K\"ahler metric with a negative lower curvature bound using the Kendall-Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete non-compact K\"ahler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau--Royden's Schwarz lemma. We also prove coupling estimates on quaternionic K\"ahler manifolds. As a byproduct, we obtain an improved gradient estimate of positive harmonic functions on K\"ahler manifolds and quaternionic K\"ahler manifolds under lower curvature bounds.