Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons

Abstract

On gradient shrinking Ricci solitons, we observe that the study of Schrödinger heat kernel seems to be more natural than the classical heat kernel. In this paper we derive sharp Gaussian upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons. As applications, we prove sharp upper bounds for the Green's function of the Schrödinger operator. We also prove sharp lower bounds for eigenvalues of the Schrödinger operator. These sharp cases are all achieved at Euclidean Gaussian shrinking Ricci solitons.

revised version, accepted by Manuscripta Math

References in corpus (1)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Cited by in corpus (2)

• Entropy and heat kernel bounds on a Ricci flow background
• Gaussian upper bounds for the heat kernel on evolving manifolds