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 Σ n − 1 \Sigma ^{n-1} embedded in H n \mathbb {H}^n with V H k VH_k being constant for some k = 1 , ⋯ , n − 1 k=1,\cdots ,n-1 is a centered geodesic sphere. Here H k H_k is the k k -th normalized mean curvature of Σ \Sigma induced from H n \mathbb {H}^n and V = cosh ⁡ r V=\cosh r , where r r is a hyperbolic distance to a fixed point in H n \mathbb {H}^n . Moreover, this result can be generalized to a compact hypersurface Σ \Sigma embedded in H n \mathbb {H}^n with the ratio V ( H k H j ) ≡ constant , 0 ≤ j > k ≤ n − 1 V\left (\frac {H_k}{H_j}\right )\equiv \mbox {constant},\;0\leq j> k\leq n-1 and H j H_j not vanishing on Σ \Sigma .