Finiteness of Small Eigenvalues of Geometrically Finite Rank one Locally Symmetric Manifolds

TL;DR

This paper proves that the Laplace operator on geometrically finite rank one locally symmetric manifolds has a finite spectrum within a small, optimal interval, advancing understanding of spectral properties in geometric analysis.

Contribution

It establishes the finiteness of small eigenvalues for a broad class of symmetric manifolds, providing an optimal bound and deepening spectral theory insights.

Findings

Finite number of small eigenvalues proven

Optimal bounds for the spectrum established

Enhanced understanding of spectral properties of symmetric manifolds

Abstract

Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.