The wave kernel on asymptotically complex hyperbolic manifolds

TL;DR

This paper investigates the wave kernel on asymptotically complex hyperbolic manifolds, showing it belongs to Fourier integral operators, analyzing its trace, and deriving an asymptotic expansion at zero time.

Contribution

It establishes the Fourier integral operator nature of the wave kernel and analyzes its trace, linking singularities to closed geodesics on such manifolds.

Findings

Wave kernel is a Fourier integral operator.

Trace singularities correspond to closed geodesic lengths.

Asymptotic expansion of the trace at zero time obtained.

Abstract

We study the behavior of the wave kernel of the Laplacian on asymptotically complex hyperbolic manifolds for finite times. We show that the wave kernel on such manifolds belongs to an appropriate class of Fourier integral operators and analyze its trace. This construction proves that the singularities of its trace are contained in the set of lengths of closed geodesics and we obtain an asymptotic expansion for the trace at time zero.