A sharp inequality relating scalar curvature, bottom spectrum, and the relative $\hat{A}$-cowaist on complete manifolds

Abstract

We introduce a refined notion of relative $\hat{A}$-cowaist, extending the framework of Cecchini--Zeidler to complete manifolds, which may be noncompact and have compact boundary. We establish a sharp inequality relating this invariant to the scalar curvature and the bottom spectrum of the Laplacian. As a consequence, we partially answer a question raised by Shi, and we obtain, in all dimensions, sharp upper bounds for the bottom spectrum under scalar curvature lower bounds. In particular, this removes the bounded geometry assumption from a result of Wang--Zhu. Our approach is based on deformed Dirac operators.

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