Long-Time Estimates for Heat Flows on Asymptotically Locally Euclidean Manifolds

We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel L^p − L^q decay estimates are established, allowing the Schrödinger operator to have a non-trivial L²-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac-type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac-type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.