Geometric flows and differential Harnack estimates for heat equations with potentials

Masashi Ishida

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


Let M be a closed Riemannian manifold with a Riemannian metric gij(t) evolving by a geometric flow ∂tgij = -2Sij, where Sij(t) is a symmetric two-tensor on (M, g(t)). Suppose that Sij satisfies the tensor inequality 2H(S, X)+E(S,X) ≥ 0 for all vector fields X on M, where H(S,X) and E(S, X) are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where Sij = Rij, the Ricci tensor of M, our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983-989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101-142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.

Original languageEnglish
Pages (from-to)287-302
Number of pages16
JournalAnnals of Global Analysis and Geometry
Issue number4
Publication statusPublished - 2014 Apr


  • Differential Harnack Estimates
  • Geometric flows
  • Heat equations with potentials
  • Ricci flow

ASJC Scopus subject areas

  • Analysis
  • Political Science and International Relations
  • Geometry and Topology


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