Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

Abstract

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our main result yields the equivalence of (i) dynamic convexity of the Boltzmann entropy on the (time-dependent) $L^2$-Wasserstein space; (ii) monotonicity of $L^2$-Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time); (iii) gradient estimates for the heat flow acting on functions (forward in time); (iv) a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI-flow for the (time-dependent) energy in $L^2$-Hilbert space and the dual heat flow on probability measures as the unique backward EVI-flow for the (time-dependent) Boltzmann entropy in $L^2$-Wasserstein space.