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Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows

Li-Juan Cheng and Anton Thalmaier

Vol. 16 (2023), No. 7, 1589–1620
Abstract

Let M be a differentiable manifold endowed with a family of complete Riemannian metrics g(t) evolving under a geometric flow over the time interval [0,T[. We give a probabilistic representation for the derivative of the corresponding conjugate semigroup on M which is generated by a Schrödinger-type operator. With the help of this derivative formula, we derive fundamental Harnack-type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.

Keywords
heat equation, geometric flow, log-Sobolev inequality, Harnack inequality, Bismut formula, Feynman–Kac formula
Mathematical Subject Classification
Primary: 35K08
Secondary: 46E35, 42B35, 35J15
Milestones
Received: 5 February 2021
Revised: 2 October 2021
Accepted: 24 January 2022
Published: 21 September 2023
Authors
Li-Juan Cheng
School of Mathematics
Hangzhou Normal University
Hangzhou
China
Anton Thalmaier
Department of Mathematics
University of Luxembourg
Esch-sur-Alzette
Luxembourg

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