Bottom spectrum and Llarull's theorem on complete noncompact manifolds

TL;DR

This paper extends Llarull's theorem to complete noncompact manifolds and those with boundary, providing bounds on scalar curvature using the bottom spectrum of the Laplacian, employing deformed Dirac operators.

Contribution

It generalizes existing theorems to broader classes of manifolds and introduces a novel approach using deformed Dirac operators for these bounds.

Findings

Extended Llarull's theorem to noncompact manifolds

Provided scalar curvature bounds via bottom spectrum

Relaxed boundary positivity conditions

Abstract

In this paper, we prove an extension of the noncompact version of Llarull's theorem due to Zhang and Li-Su-Wang-Zhang, giving an upper bound for the infimum of scalar curvature in terms of the bottom spectrum of the Laplacian. Moreover, we extend the theorem to manifolds with boundary, relaxing the strict positivity condition on the scalar curvature near the boundary that was required by Liu-Liu. Our approach is based on deformed Dirac operators.