Sharp bottom spectrum and scalar curvature rigidity

TL;DR

This paper derives a precise upper bound for the lowest eigenvalue of the Beltrami Laplacian on universal covers of closed manifolds with scalar curvature constraints and proves a rigidity theorem when this bound is attained.

Contribution

It introduces a sharp spectral bound linked to scalar curvature and establishes a rigidity result, extending scalar curvature characterizations to noncompact manifolds.

Findings

Established a sharp upper bound for the bottom spectrum of the Beltrami Laplacian.

Proved a scalar curvature rigidity theorem when the spectral bound is achieved.

Provided a net characterization of scalar curvature for complete noncompact manifolds.

Abstract

We establish a sharp upper bound for the bottom spectrum of the Beltrami Laplacian on universal covers of closed Riemannian manifolds with scalar curvature lower bound. Moreover, we prove a scalar curvature rigidity theorem when this bound is achieved. Additionally, we prove a net characterization of scalar curvature for general complete noncompact Riemannian manifolds.