The lower bound of the Ricci curvature that yields the infinite number of the discrete spectrum of the Laplacian

TL;DR

This paper investigates the precise curvature conditions under which the Laplace-Beltrami operator on a manifold has an infinite discrete spectrum, establishing a clear boundary for this spectral property.

Contribution

It provides a complete characterization of the curvature bounds that determine whether the Laplacian's discrete spectrum is infinite or finite.

Findings

Identifies the exact Ricci curvature lower bounds for infinite discrete spectrum

Establishes a sharp criterion for the finiteness or infiniteness of the spectrum

Clarifies the borderline behavior of curvature related to spectral properties

Abstract

This paper discusses the question whether the discrete spectrum of the Laplace-Beltrami operator is infinite or finite. The borderline-behavior of the curvatures for this problem will be completely determined.