Weyl law for the volume spectrum

TL;DR

This paper proves a Weyl law for the volume spectrum of Riemannian manifolds, confirming a conjecture by Gromov, and providing asymptotic behavior of the spectrum.

Contribution

It establishes the Weyl law for the volume spectrum of Riemannian manifolds, a significant step in geometric analysis.

Findings

Confirmed Gromov's conjecture on the Weyl law for volume spectrum

Derived asymptotic formulas for the volume spectrum

Extended results to manifolds with boundary

Abstract

Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.