Perelman's W-entropie for the Witten Laplacian and for the Fokker-Planck equation on Riemannian manifolds
by Xiangdong Li (Chinese Academy of Sciences)
In 2002, G. Perelman introduced the mysterious W-entropy for Ricci flow and proved its monotonicity along the conjugate heat equation. In this talk, we study the Perelman W-entropy for the heat equation of the Witten Laplacian and for the Fokker-Planck equation on complete Riemannian manifolds. Under some natural geometric condition in terms of dimension and the Bakry-Emery Ricci curvature, we prove the monotonicity and the rigidity theorems of the W-entropy for these equations.