Local Limit Theorem in negative curvature

TL;DR

This paper proves a local limit theorem for the heat kernel on negatively curved manifolds, characterizes the Martin boundary, and explores associated measure families using advanced geometric and dynamical tools.

Contribution

It establishes the local limit theorem for the heat kernel on negatively curved spaces and identifies the Martin boundary with the topological boundary, including measure decomposition and uniqueness results.

Findings

Proved the local limit theorem for the heat kernel in negative curvature.

Identified the Martin boundary with the topological boundary.

Characterized the measure family as energy-minimizing and unique.

Abstract

Consider the heat kernel $p(t,x,y)$ on the universal cover $X$ of a Riemannian manifold $M$ of negative curvature. We show the local limit theorem for $p$ : $$\lim_{t \to \infty} t^{3/2}e^{\lambda_0 t} p(t,x,y)=C(x,y),$$ where $\lambda_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive function which depends on $x, y \in X$. We also show that the $\lambda_0$-Martin boundary of $X$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{\mu^{\lambda_0}_x \}$ on $\partial \widetilde{M}$. We show that $\{\mu^{\lambda_0}_x \}$ is the unique family minimizing the energy or the Rayleigh quotient of Mohsen. We use the uniform Harnack inequality on the boundary $\partial X$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs-Margulis measures.