On the spectral geometry of Liouville quantum gravity

TL;DR

This paper explores the spectral properties of Liouville quantum gravity, including eigenvalues, eigenfunctions, and the Weyl law, advancing understanding of diffusion processes in this complex geometric setting.

Contribution

It provides a concise construction of the LQG spectrum, summarizes recent Weyl law results, and discusses open problems in the spectral geometry of LQG.

Findings

Construction of LQG eigenvalues and eigenfunctions

Summary of the Weyl law in LQG context

Discussion of open problems and heat trace work

Abstract

We give a concise presentation of the construction of the Liouville quantum gravity (LQG) eigenvalues and eigenfunctions, i.e., the spectrum associated to the infinitesimal generator of Liouville Brownian motion, the canonical diffusion in the geometry of LQG. We describe the recently obtained Weyl law in this context (giving a short summary of its proof) and report on some work in progress concerning the associated heat trace. Finally, we summarise and propose some new key open problems in this direction.