Heat Determinant on Manifolds

TL;DR

This paper introduces new geometric invariants derived from the heat kernel on manifolds, which depend on both eigenvalues and eigenfunctions, providing deeper insights into the manifold's structure.

Contribution

It presents the construction and explicit computation of novel heat kernel invariants that incorporate eigenfunction information, extending traditional spectral invariants.

Findings

Explicit formulas for the first three low-order invariants

Demonstration that invariants depend on eigenfunctions beyond eigenvalues

New tools for analyzing manifold geometry using heat kernel invariants

Abstract

We introduce and study new invariants associated with Laplace type elliptic partial differential operators on manifolds. These invariants are constructed by using the off-diagonal heat kernel; they are not pure spectral invariants, that is, they depend not only on the eigenvalues but also on the corresponding eigenfunctions in a non-trivial way. We compute the first three low-order invariants explicitly.