W-entropy formula on super Ricci flow and optimal transportation on manifolds, from Perelman to Lott-Villani
主 题: W-entropy formula on super Ricci flow and optimal transportation on manifolds, from Perelman to Lott-Villani
报告人: 李向东教授 (中科院应用数学研究所)
时 间: 2014-03-04 15：00-16：00
地 点: 理科一号楼 1114(数学所活动）
To complete Hamilton's project for the proof of the Poincare conjecture, G. Perelman interpreted the Ricci flow as the gradient flow of the so-called F-entropy functional, and proved the W-entropy formula along the associated conjugate heat equation. Inspired by Perelman's work, we prove the W-entropy formula for the heat equation of the time dependent Witten Laplacian on manifolds equipped with super Ricci flow. We then prove the W-entropy formula for the transport equation and the Hamilton-Jacobi equation on manifolds. Our work recaptures Lott-Villani's theorem on the displacement convexity of the Boltzmann entropy on the Wasserstein space over Riemannian manifolds with non-negative Ricci curvature, an important result in the topic of the optimal transportation problem on manifolds. To better understand the above two results, we introduce the Langevin deformation of flows on the Wasserstein space, which interpolates the heat equation on manifolds and the geodesic flow on the cotangent bundle over the Wasserstein space, and can be considered as the potential flow of the compressible Euler equation with damping on Riemannian manifolds. The W-entropy formula will be extended to the deformation flows and the rigidity model is proposed. Joint work with Songzi Li (Fudan University and University Paul Sabatier).