# Long-time estimates for heat flows on ALE manifolds

#### 10.12.2020, 16:15  –  Online-Seminar Forschungsseminar Differentialgeometrie

We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Assuming that the Schrödinger operator can be written as the square of a Dirac-type operator, we prove new $L^p-L^q$ estimates for the heat kernel and its derivatives. The purpose of this assumption is to apply the Fredholm theory for first order elliptic operators on ALE manifolds. If the ALE manifold carries a parallel spinor, then a result by Wang says that the Lichnerowicz Laplacian can be written as a square of a twisted Dirac operator. Hence our result is relevant for Ricci flow. This is joint work with Klaus Kröncke.