DIFFERENTIAL HARNACK AND LOGARITHMIC SOBOLEV INEQUALITIES ALONG RICCI-HARMONIC MAP FLOW

This paper introduces a new family of entropy functionals which is proved to be monotonically nondecreasing along the Ricci-harmonic map heat flow. Some of the consequences of the monotonicity are combined to derive gradient estimates and Harnack inequalities for all positive solutions to the associated conjugate heat equation. We relate the entropy monotonicity and the ultracontractivity property of the heat semigroup, and as a result we obtain the equivalence of logarithmic Sobolev inequalities, conjugate heat kernel upper bounds and uniform Sobolev inequalities under Ricci-harmonic map heat flow.