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Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

Abstract

In this paper, we study Perelman’s \({{\mathcal W}}\) -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the \({\mathcal{W}}\) -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the \({{\mathcal W}}\) -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the \({\mathcal{W}}\) -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.

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Correspondence to Xiang-Dong Li.

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Dedicated to the memory of my advisor: Professor Paul Malliavin.

X.-D. Li’s research was supported by NSFC No. 10971032, Shanghai Pujiang Talent Project No. 09PJ1401600, Key Laboratory RCSDS, CAS, No. 2008DP173182, and a Hundred Talents Project of CAS.

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Li, XD. Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature. Math. Ann. 353, 403–437 (2012). https://doi.org/10.1007/s00208-011-0691-y

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Keywords

  • Riemannian Manifold
  • Heat Equation
  • Heat Kernel
  • Ricci Curvature
  • Compact Riemannian Manifold