A new characterization of geodesic spheres in the hyperbolic space

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 $\mathbb {H}^n$ 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 $\mathbb {H}^n$ and $V=\cosh r$, where $r$ is a hyperbolic distance to a fixed point in $\mathbb {H}^n$. Moreover, this result can be generalized to a compact hypersurface $\Sigma$ embedded in $\mathbb {H}^n$ 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 $\Sigma$.