Yamabe flow and metrics of constant scalar curvature on a complete manifold



In this paper, we give a condition on the initial metric which makes the global existence of Yamabe flow and we study the global behavior of the Yamabe flow in a complete noncompact Riemannian manifold. As application we study the ADM mass monotonicity of Yamabe flow in AF manifolds. We use the variational method to study the existence problem of metrics with constant scalar curvature on complete non-compact Riemannian manifolds and we can give a partial affirmative answer to a question posed by Kazdan (Math Ann 261(2):227–234, 1982).

Mathematics Subject Classification

53C20 35Jxx 58J20 



The author is very grateful to the unknown referees for helpful suggestions, in particular for pointing out the reference [32]. Part of the revision had been done when the author visited the Department of Mathematics at Stanford University in June and July of 2018 and the author would like to thank Prof. R. Schoen for the invitation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.Department of MathematicsHenan Normal UniversityXinxiangPeople’s Republic of China

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