Bottom spectrum, vertical $\widehat{A}$-cowaist and scalar curvature rigidity

TL;DR

This paper introduces the vertical -cowaist invariant for partitioned manifolds and establishes inequalities linking scalar curvature and spectral properties, extending previous results to more general settings.

Contribution

It extends the concept of vertical -cowaist to arbitrary partitioned manifolds and proves new inequalities and theorems using deformed Dirac operators.

Findings

Established a sharp inequality relating scalar curvature, bottom spectrum, and the -cowaist.

Derived a high-dimensional analogue of Munteanu-Wang's bottom spectrum estimate.

Proved a quantitative strengthening of Anghel's theorem and a boundary version.

Abstract

We introduce the vertical \(\widehat{A}\)-cowaist, a codimension-one invariant for partitioned manifolds. It extends the concept of infinite vertical \(\widehat{A}\)-cowaist for bands to arbitrary partitioned manifolds, which may be noncompact and have compact boundary. We establish a sharp inequality relating the scalar curvature, the bottom spectrum of the Laplacian, and this invariant. As an application, we obtain a high-dimensional analogue of Munteanu-Wang's bottom spectrum estimate. We also prove a quantitative strengthening of Anghel's theorem together with a boundary version, as well as a Calabi-Yau type theorem that goes beyond the dimensional restrictions of the earlier \(\mu\)-bubble method. Our approach is based on deformed Dirac operators.