Heat Kernel Upper Bound on Riemannian Manifolds with Locally Uniform Ricci Curvature Integral Bounds

Abstract

This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature the heat kernel admits a Gaussian upper bound for small times. This provides general assumptions on the geometry of a manifold such that certain function spaces are in the Kato class. Additionally, the results imply bounds on the first Betti number.

Geometric inequalities for manifolds with Ricci curvature in the Kato class

“…Recents papers have emphasized how a control on the Kato constant of the Ricci curvature can be useful in order to control some geometrical quantities for closed or complete Riemannian manifolds ( [5,17,20,25,47,48,49,58,59]). For a closed Riemannian manifold (M, g), we will explain how the works of Qi S. Zhang and M. Zhu [59], together with some classical ideas, can be used in order to obtain geometric and topological estimates based on the Kato constant of the Ricci curvature.…”

Heat Kernel Approximation on Kendall Shape Space

“…Consequently, the heat kernel is appropriate for constructing nonlinear support vector machines for shapes' classification [43] . It is worth noting that to better grasp the significance of the heat kernel closed-form approximation in (11) it should be reminded that the straightforward resolution of heat equations is only feasible for a restricted set of classical manifolds [24][25][26][27] ; other approaches provide merely bounds of the heat kernel [28][29][30][31][32][33][34][35][36][37][38] . Furthermore, number of closed-form expressions have been proposed in the litterature to calculate the heat kernel on hyperspheres like k m S .…”

Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

“…Furthermore, it was shown that under suitable conditions, the Hodge-deRham Laplacian is dominated by a Schrödinger operator generated by the Laplace-Beltrami plus a suitable potential depending on Ricci curvature, as we will use later. Especially in Riemannian geometry, this fact has been used extensively to study geometric and topological properties of manifolds as well as properties of the semigroup and corresponding heat kernel of generalized Schrödinger operators on vector bundles, see [5,9,[12][13][14][22][23][24][25] and the references therein. For a recent survey, see Sect.…”

“…We conclude the paper with the following list of examples, stating conditions on M where quantitative heat kernel bounds can be obtained. For more details, the reader should consult [3,11,22,24,26].…”