In this paper, we develop a new approach to prove the W -entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the W -entropy formula. Then we prove the W -entropy formula for the Witten Laplacian on compact Riemannian manifolds with time dependent metrics and potentials, and derive the W -entropy formula for the backward heat equation associated with the Witten Laplacian on compact Riemannian manifolds equipped with Lott's modified Ricci flow. We also extend our results to complete Riemannian manifolds with negative m-dimensional Bakry-Emery Ricci curvature, and to compact Riemannian manifolds with K-super m-dimensional Bakry-Emery Ricci flow. As application, we prove that the optimal logarithmic Sobolev constant on compact manifolds equipped with the K-super mdimensional Bakry-Emery Ricci flow is decreasing in time.

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation ∂ t u = Lu associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the (K, m)-super Perelman Ricci flows and the K-super Perelman Ricci flows.

In this survey paper, we give an overview of our recent works on the study of the W -entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W -entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W -entropy formula for the heat equation associated with the time dependent Witten Laplacian on n-dimensional compact manifolds equipped with a (K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the Wentropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W -entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the W -entropy formula for the Langevin deformation. Finally, we make a discussion on the W -entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.MSC2010 Classification: primary 58J35, 58J65; secondary 60J60, 60H30.It is natural and interesting to ask the problems what is the hidden idea for Perelman to introduce the mysterious W -entropy, what is the reason for him to call the quantity in (1) the W -entropy, and whether there is some essential link between the W -entropy and the Boltzmann entropy in statistical mechanics and probability theory.Inspired by Perelman [44] and related works [39,40], the second author of this paper proved in [20] the W -entropy formula for the heat equation of the Witten Laplacian on compact Riemannian manifolds with the CD(0, m)-condition and gave a probabilistic interpretation of the W -entropy for the Ricci flow. Later, the W -entropy formula and a rigidity theorem for the W -entropy were proved in [22,24] for the fundamental solution to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(0, m)condition, and the W -entropy formula was proved in [21] for the Fokker-Planck equation of the Witten Laplacian on complete Riemannian manifolds with the CD(0, m)-condition. The relationship between Perelman's W -entropy formula for the Ricci flow and the Boltzmann H-theorem for the Boltzmann equation was discussed in [23] from the point of view of the statistical mechanics. In [26,27,29,30,31], we extended the W -entropy formula to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)condition and on compact Riemannian manifolds equipped with (K, m)-super Ricci flows. Moreover, we proved in [28,31] an analogue of the W -entropy formula for the geodesic flow on the Was...

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Abstract. In this paper, we prove the Hamilton differential MSC2010 Classification: primary 53C44, 58J35, 58J65; secondary 60J60, 60H30.