J. Korean Math. Soc. 2018; 55(3): 735-747
Online first article December 12, 2017 Printed May 1, 2018
https://doi.org/10.4134/JKMS.j170440
Copyright © The Korean Mathematical Society.
Xiaomin Chen
China University of Petroleum-Beijing
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 $c\neq0$ 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.
Keywords: Miao-Tam critical metric, Hopf hypersurface, non-flat complex space form, ruled hypersurface, complex Euclidean space
MSC numbers: Primary 53C25, 53D10
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