• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.735-747
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• 2018

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.