# Upper bounds for the spectral function on homogeneous spaces via volume growth

## TL;DR

This paper derives explicit upper bounds for the spectral function and heat kernel on homogeneous spaces using volume growth, extending and improving classical eigenvalue bounds for compact manifolds.

## Contribution

It introduces new upper bounds for the spectral function on homogeneous spaces based on volume growth, extending Li's eigenvalue bounds to all eigenvalues and improving the original bounds.

## Key findings

- Upper bounds for spectral functions in terms of volume growth
- Extension of Li's eigenvalue bounds to all eigenvalues
- Explicit heat kernel bounds for homogeneous spaces

## Abstract

We use spectral embeddings to give upper bounds on the spectral function of the Laplace--Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also improve Li's bound itself. Our bounds translate to explicit upper bounds on the heat kernel for both compact and noncompact homogeneous spaces.