Brownian Motions and Heat Kernel Lower Bounds on Kähler and Quaternion Kähler Manifolds

Abstract

We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger–Yau-type lower bound for the heat kernels of such manifolds and also sharp Cheng’s type estimates for the Dirichlet eigenvalues of metric balls.

A note on first eigenvalue estimates by coupling methods in Kähler and quaternion Kähler manifolds

“…In this section, for the sake of completeness, we give the definitions we will be using in this paper. We refer to [1] for more details. Throughout the paper, let (M, g) be a smooth complete Riemannian manifold.…”

“…Definition 2.1. The manifold (M, g) is called a Kähler manifold, if there exists a smooth (1,1) tensor J on M that satisfies:…”

“…where Ric is the usual Riemannian Ricci tensor of (M, g). The table below shows the curvature of the quaternion-Kähler model spaces H m , HP m and HH m , see [1].…”

“…Remark 4.2. Theorem 4.1 can be used to improve the lower bound of Theorem 1.1 in some situations like for example the complex projective space CP m in Table (1). Indeed H ≥ 4k by concavity of k → G(k, r).…”

“…It seems worthy to mention that to obtain Laplacian and Index comparison theorems with model spaces in Kähler (and quaternion Kähler) geometry, it is necessary that the lower bounds of orthogonal Ricci curvature and holomorphic (quaternionic) sectional curvature should be assumed simultaneously, not only a Ricci curvature lower bound (also, see [5]). In this paper, the main geometric ingredients are new estimates of the index form in those settings that build on the previous recent work [1].…”