Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension.

Gromov conjectured that the scalar curvature on uniformly contractible complete Riemannian manifolds cannot be uniformly positive. This conjecture holds for manifolds with finite asymptotic dimension. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. This result is inspired by the quadratic decay inequality of scalar curvature for a special class of complete manifolds of Gromov and Zeidler. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topological $K$-theory of finite dimensional simplicial complexes. We also construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly.