Differential Harnack and logarithmic Sobolev inequalities along Ricci-harmonic map flow (Q745091)

Let \((M^n, g)\) and \((N^m, h)\) be compact Riemannian manifolds and let \(u : M \to N\) be a harmonic map. The pair \((g(x,t), u(x,t)), t \in [0, T)\), of a one-parameter family of Riemannian metrics \(g(x,t)\) and a family of smooth maps \(u(x, t)\) is defined to be a Ricci-harmonic map heat flow if it satisfies the coupled system of nonlinear parabolic equations defined by \[ \begin{cases} \frac{\partial}{\partial t} g(x,t) = - 2\text{Ric}(x,t) + 2\alpha \nabla u(x,t) \otimes \nabla u(x,t),\\ \frac{\partial}{\partial t} u(x,t) = \tau_g u(x,t), \end{cases} \] where \(\alpha(t) >0\) is a time-dependent constant and \(\tau_g u(x,t)\) is the tension field. This system was first studied by \textit{B. List} [Commun. Anal. Geom. 16, No. 5, 1007--1048 (2008; Zbl 1166.53044)] in a special case \(N = {\mathbb R}\) and \(\alpha =2\), and generalized by \textit{R. Müller} [Ann. Sci. Éc. Norm. Supér. (4) 45, No. 1, 101--142 (2012; Zbl 1247.53082)] to the general case. In this paper, the author studies the behavior of all positive solutions to the associated conjugate heat equation along the Ricci-harmonic map heat flow. First he introduces a new family of entropy functionals which generalize Perelman and Müller's \({\mathcal{W}}_\alpha\)-entropy, and proves monotonicity formula along the Ricci-harmonic map heat flow. Then using the entropy monotonicity, the author derives gradient estimates and differential Harnack inequalities for all positive solutions to the associated conjugate heat equation. The author also obtains the equivalence of logarithmic Sobolev inequalities, conjugate heat kernel upper bounds and uniform Sobolev inequalities under Ricci-harmonic map heat flow.