A sharp inequality between scalar curvature and the bottom spectrum on complete manifolds

TL;DR

This paper establishes a precise inequality connecting scalar curvature and the bottom spectrum on complete manifolds, using deformed Dirac operators, with implications for geometric conjectures and spectral estimates.

Contribution

It generalizes the relative ermann- Zeidler A-cowaist concept and derives a sharp inequality linking scalar curvature and spectral properties.

Findings

Proves a sharp inequality between scalar curvature and bottom spectrum.

Provides estimates for the bottom spectrum under scalar curvature bounds.

Advances the understanding of the Geroch conjecture and related geometric problems.

Abstract

In this paper, we generalize the notion of relative $\widehat{A}$-cowaist, introduced by Cecchini and Zeidler, and establish a sharp inequality linking it to scalar curvature and the bottom spectrum. This yields a number of geometric applications, including progress on the generalized Geroch conjecture and estimates for the bottom spectrum under scalar curvature lower bounds. Our approach is based on deformed Dirac operators.