Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound

TL;DR

This paper extends Cheng's classical result by establishing the maximal bottom spectrum for three-dimensional complete manifolds with scalar curvature lower bounds, under certain topological conditions, and proves a related splitting theorem.

Contribution

It provides a new spectral characterization for 3D manifolds with scalar curvature bounds and addresses rigidity through a splitting theorem, filling a gap in geometric analysis.

Findings

Maximal bottom spectrum achieved by certain 3D manifolds

Rigidity results for manifolds with maximal spectrum

Splitting theorem for manifolds with maximal bottom spectrum

Abstract

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum.