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arXiv · 2006.06662

Long-time estimates for heat flows on ALE manifolds

Klaus Kroencke·Oliver Lindblad Petersen
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Abstract

We consider the heat equation associated to Schr\"{o}dinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel $L^p - L^q$ decay estimates are established, allowing the Schr\"{o}dinger operator to have a non-trivial $L^2$-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 Schr\"{o}dinger 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.

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Klaus Kroencke, Oliver Lindblad Petersen. 2020-06-11. Long-time estimates for heat flows on ALE manifolds. https://doi.org/10.1093/imrn%2Frnab350

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