论文标题
耦合及其应用的Kähler歧管上的随机Schwarz引理
The Stochastic Schwarz lemma on Kähler Manifolds by Couplings and Its Applications
论文作者
论文摘要
我们首先根据一般的马尔可夫耦合提供了Carathéodory距离的随机公式，并证明了Carathéodory距离与完整的Kähler指标之间的比较结果，其使用Kendall-Cranston耦合较低的曲率较低。这种概率方法给出了完全非紧凑的kähler歧管上的Schwarz引理版本，并进一步分解RICCI曲率为正交的RICCI曲率和全体形态截面曲率，无法通过使用Yau-yau-yau-yau-yau-ryroyden的Schwarz Lemma来获得。我们还证明了关于QuaternionicKähler歧管的耦合估计。作为副产品，我们在较低的曲率界限下获得了Kähler歧管和QuaternionicKähler歧管上正谐波功能的梯度估计。
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 non-compact 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 byproduct, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.
