The heat equation and independence of the spectrum of the Hodge Laplacian on $\ell^p$

TL;DR

This paper investigates the heat equation related to the Hodge Laplacian on simplicial complexes, establishing kernel estimates, extending semigroup properties across different p-norms, and proving spectrum independence under certain geometric conditions.

Contribution

It introduces new heat kernel estimates and extends the spectral analysis of the Hodge Laplacian to various p-spaces, demonstrating spectrum independence under specific curvature and volume growth assumptions.

Findings

Established Davies-Gaffney-Grigoryan type estimates for the heat kernel.

Extended the heat semigroup to all p in [1, ∞] under geometric conditions.

Proved p-independence of the Hodge Laplacian spectrum with curvature and volume growth assumptions.

Abstract

We study the heat equation associated to the Hodge Laplacian on simplicial complexes. Using recently developed techniques for magnetic Schr\"odinger operators, we prove Davies-Gaffney-Grigoryan type estimates for the kernel of the heat semigroup on $\ell^2,$ which we then use to extend the semigroup to $\ell^p$ for $p\in[1,\infty]$ under suitable curvature and volume growth conditions. Furthermore, we establish $p$-independence of the Hodge Laplacian spectrum under the assumption of form bounded curvature and uniform subexponential volume growth. While the main focus of the paper is the Hodge Laplacian on simplicial complexes, the results are indeed proven for general positive magnetic Schr\"odinger operators on graphs.