Heat kernel estimates based on Ricci curvature integral bounds

Más información sobre el libro

Any Riemannian manifold has a minimal solution to the heat equation for the Dirichlet Laplacian, known as the heat kernel. Recent decades have seen extensive investigation into the geometric properties of manifolds that allow their heat kernels to satisfy Gaussian upper bounds. Researchers have particularly focused on compact and non-compact manifolds with lower bounded Ricci curvature, leading to Gaussian estimates, especially in compact cases with integral Ricci curvature assumptions. Key techniques for achieving these bounds include the symmetrization procedure for compact manifolds and relative Faber-Krahn estimates, which rely on isoperimetric properties. This thesis extends existing results by demonstrating that locally uniform integral bounds on the negative part of Ricci curvature yield Gaussian upper bounds for the heat kernel, regardless of compactness. We establish local isoperimetric inequalities and apply relative Faber-Krahn estimates to derive explicit Gaussian upper bounds. For compact manifolds, we can extend the integral curvature condition to cases where the negative part of Ricci curvature lies in the Kato class, achieving uniform Gaussian upper bounds through gradient estimate techniques. Additionally, we utilize these estimates to generalize Bochner’s theorem, leading to ultracontractive estimates for the heat semigroup and the Hodge Laplacian. This allows us to formulate rigidity results regarding the tr