A new characterization of geodesic spheres in the Hyperbolic space

TL;DR

This paper introduces a new way to characterize geodesic spheres in hyperbolic space using a weighted higher order mean curvature, generalizing previous results and providing conditions for identifying such spheres.

Contribution

It presents a novel characterization of geodesic spheres in hyperbolic space based on weighted higher order mean curvature conditions, extending existing geometric criteria.

Findings

Hypersurfaces with constant weighted higher order mean curvature are geodesic spheres.

The characterization extends to hypersurfaces with constant ratios of weighted mean curvatures.

The results generalize classical sphere characterization theorems in hyperbolic geometry.

Abstract

This paper gives a new characterization of geodesic spheres in the hyperbolic space in terms of a ``weighted'' higher order mean curvature. Precisely, we show that a compact hypersurface $\Sigma^{n-1}$ embedded in $\H^n$ with $VH_k$ being constant for some $k=1,\cdots,n-1$ is a centered geodesic sphere. Here $H_k$ is the $k$-th normalized mean curvature of $\Sigma$ induced from $\H^n$ and $V=\cosh r$, where $r$ is a hyperbolic distance to a fixed point in $\H^n$. Moreover, this result can be generalized to a compact hypersurface $\Sigma$ embedded in $\H^n$ with the ratio $V\left(\frac{H_k}{H_j}\right)\equiv\mbox{constant},\;0\leq j< k\leq n-1$ and $H_j$ not vanishing on $\Sigma$.