Dimension-free Harnack inequalities for conjugate heat
equations and their applications to geometric flows
by Li-Juan Cheng and Anton Thalmaier
Abstract
Let be a differentiable manifold endowed with a family of
complete Riemannian metrics evolving under a geometric flow
over the time interval . In this article, we give a
probabilistic representation for the derivative of the
corresponding conjugate semigroup on 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.
Analysis & PDE 16 (2023), no. 7, 1589-1620
The paper is available here:
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