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Title: The stochastic Schwarz lemma on Kähler manifolds by couplings and its applications
Abstract

We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall–Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete noncompact Kähler 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 Kähler manifolds. As a by‐product, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.

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Award ID(s):
1954264
NSF-PAR ID:
10482259
Author(s) / Creator(s):
 ;  ;  ;  
Publisher / Repository:
Oxford University Press (OUP)
Date Published:
Journal Name:
Journal of the London Mathematical Society
Volume:
109
Issue:
1
ISSN:
0024-6107
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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