Heat Kernel on Warped Products

TL;DR

This paper analyzes the spectral properties of the Laplacian on warped product manifolds, deriving explicit formulas for the heat kernel, resolvent, and scattering matrix, and exploring their asymptotic behaviors in both compact and non-compact cases.

Contribution

It provides a detailed spectral analysis of the Laplacian on warped product manifolds, including explicit computations of the heat kernel, resolvent, and scattering matrix, with new insights into their asymptotics.

Findings

The spectrum includes both discrete and continuous parts.

Explicit formulas for the heat kernel and resolvent are derived.

Asymptotic coefficients relate to the zeta function of the base manifold.

Abstract

We study the spectral properties of the scalar Laplacian on a $n$-dimen\-sional warped product manifold $M=\Sigma\times_f N$ with a $(n-1)$-dimensional compact manifold $N$ without boundary, a one dimensional manifold $\Sigma$ without boundary and a warping function $f\in C^\infty(\Sigma)$. We consider two cases: $\Sigma=S^1$ when the manifold $M$ is compact, and $\Sigma=\mathbb{R}$ when the manifold $M$ is non-compact. In the latter case we assume that the warping function $f$ is such that the manifold $M$ has two cusps with a finite volume. In particular, we study the case of the warping function $f(y)=[\cosh(y/b)]^{-2\nu/(n-1)}$ in detail, where $y\in\mathbb{R}$ and $b$ and $\nu$ are some positive parameters. We study the properties of the spectrum of the Laplacian in detail and show that it has both the discrete and the continuous spectrum. We compute the resolvent, the eigenvalues, the scattering matrix, the heat kernel and the regularized heat trace. We compute the asymptotics of the regularized heat trace of the Laplacian on the warped manifold $M$ and show that some of its coefficients are global in nature expressed in terms of the zeta function on the manifold $N$.