Sharp Li–Yau-Type Gradient Estimates on Hyperbolic Spaces

  • Chengjie YuEmail author
  • Feifei Zhao


In this paper, motivated by the works of Bakry et al. in finding sharp Li–Yau-type gradient estimates for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li–Yau-type gradient estimate and show that the validity of such an estimate for any positive solutions of the heat equation reduces to the validity of the estimate for the heat kernel of the Riemannian manifold. Then, a sharp Li–Yau-type gradient estimate on the three-dimensional hyperbolic space is obtained by using the explicit expression of the heat kernel, and some optimal Li–Yau-type gradient estimates on general hyperbolic spaces are obtained.


Heat equation Li–Yau-type gradient estimate Heat kernel 

Mathematics Subject Classification

Primary 35K05 Secondary 53C44 



The authors would like to thank the referee for helpful comments and suggestions.


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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsShantou UniversityShantouChina

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