The Heat Kernel on Asymptotically Hyperbolic Manifolds

TL;DR

This paper provides comprehensive heat kernel bounds on asymptotically hyperbolic manifolds, showing they are comparable to hyperbolic space heat kernels for all times and distances, under specific spectral assumptions.

Contribution

It extends heat kernel estimates to all times and distances on asymptotically hyperbolic manifolds, using microlocal analysis and resolvent techniques.

Findings

Heat kernel on such manifolds is comparable to hyperbolic space kernel.

Uniform bounds hold for all times and geodesic distances.

Results depend on spectral assumptions of no eigenvalues or resonance.

Abstract

Upper and lower bounds on the heat kernel on complete Riemannian manifolds were obtained in a series of pioneering works due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. However, these estimates do not give a complete picture of the heat kernel for all times and all pairs of points. Inspired by the work of Davies-Mandouvalos on $\mathbb{H}^{n + 1}$, we study heat kernel bounds on Cartan-Hadamard manifolds that are asymptotically hyperbolic in the sense of Mazzeo-Melrose. Under the assumption of no eigenvalues and no resonance at the bottom of the continuous spectrum, we show that the heat kernel on such manifolds is comparable to the heat kernel on hyperbolic space of the same dimension (expressed as a function of time $t$ and geodesic distance $r$), uniformly for all $t \in (0, \infty)$ and all $r \in [0, \infty)$. Our approach is microlocal and based on the resolvent on asymptotically hyperbolic manifolds, constructed in the celebrated work of Mazzeo-Melrose, as well as its high energy asymptotic, due to Melrose-S\`{a} Barreto-Vasy.