Title: $W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons,
Abstract: In this paper, we prove the characterization of the $(K, \infty)$-super
Perelman Ricci flows by various functional inequalities and gradient estimate
for the heat semigroup generated by the Witten Laplacian on manifolds equipped
with time dependent metrics and potentials. As a byproduct, we derive the
Hamilton type dimension free Harnack inequality on manifolds with $(K,
\infty)$-super Perelman Ricci flows. Based on a new second order differential
inequality on the Boltzmann-Shannon entropy for the heat equation of the Witten
Laplacian, we introduce a new $W$-entropy quantity and prove its monotonicity
for the heat equation of the Witten Laplacian on complete Riemannian manifolds
with the $CD(K, \infty)$-condition and on compact manifolds with $(K,
\infty)$-super Perelman Ricci flows. Our results characterize the $(K,
\infty)$-Ricci solitons and the $(K, \infty)$-Perelman Ricci flows. We also
prove a second order differential entropy inequality on $(K, m)$-super Ricci
flows, which can be used to characterize the $(K, m)$-Ricci solitons and the
$(K, m)$-Ricci flows. Finally, we give a probabilistic interpretation of the
$W$-entropy for the heat equation of the Witten Laplacian on manifolds with the
$CD(K, m)$-condition.