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

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.