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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.
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참고문헌 (18)
참고문헌 신청
R. Batista / 2017 / Critical metrics of the volume functional on compact three-manifolds with smooth boundary / J. Geom. Anal. 27(2):1530~1547
J. Berndt / 1989 / Real hypersurfaces with constant principal curvatures in complex hyperbolic space / J. Reine Angew. Math. 395:132~141
T. E. Cecil / 1982 / Focal sets and real hypersurfaces in complex projective space / Trans. Amer. Math. Soc. 269(2):481~499
J. T. Cho / 2009 / Ricci solitons and real hypersurfaces in a complex space form / Tohoku Math. J. 61(2):205~212
J. T. Cho / 2011 / Ricci solitons of compact real hypersurfaces in Kahler manifolds / Math. Nachr. 284(11-12):1385~1393
J. Berndt / 1989 / Real hypersurfaces with constant principal curvatures in complex hyperbolic space / J. Reine Angew. Math. 395:132~141
T. E. Cecil / 1982 / Focal sets and real hypersurfaces in complex projective space / Trans. Amer. Math. Soc. 269(2):481~499
J. T. Cho / 2009 / Ricci solitons and real hypersurfaces in a complex space form / Tohoku Math. J. 61(2):205~212
J. T. Cho / 2011 / Ricci solitons of compact real hypersurfaces in Kahler manifolds / Math. Nachr. 284(11-12):1385~1393
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