Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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REAL HYPERSURFACES WITH MIAO-TAM CRITICAL METRICS OF COMPLEX SPACE FORMS

  • Chen, Xiaomin (College of Science China University of Petroleum-Beijing)
  • Received : 2017.06.29
  • Accepted : 2017.10.31
  • Published : 2018.05.01
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Abstract

Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e., the induced metric g on M satisfies the equation: $-({\Delta}_g{\lambda})g+{\nabla}^2_g{\lambda}-{\lambda}Ric=g$, where ${\lambda}$ is a smooth function on M. At first, for the case where M is Hopf, c = 0 and $c{\neq}0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space admitting Miao-Tam critical metric with ${\lambda}$ > 0 or ${\lambda}$ < 0 is a sphere and a compact hypersurface of a non-flat complex space form does not exist such a critical metric.

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References

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