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.
