Real hypersurfaces with Miao-Tam critical metrics of complex space forms

TL;DR

This paper classifies real hypersurfaces in complex space forms that admit Miao-Tam critical metrics, showing that only spheres in Euclidean space qualify among compact hypersurfaces, and non-Hopf or non-flat cases do not admit such metrics.

Contribution

It provides a complete classification of real hypersurfaces with Miao-Tam critical metrics in complex space forms, including non-existence results and characterizations of spheres.

Findings

Compact hypersurfaces with Miao-Tam critical metrics in Euclidean space are spheres.

Ruled hypersurfaces in non-flat complex space forms do not admit Miao-Tam critical metrics.

Non-existence of such metrics on non-Hopf and non-flat hypersurfaces.

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 $c\neq0$ 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.