Sphere-foliated minimal and constant mean curvature hypersurfaces in product spaces

TL;DR

This paper proves that certain minimal and constant mean curvature hypersurfaces in hyperbolic product spaces, foliated by spheres, must be rotationally symmetric, revealing a symmetry property under specific geometric conditions.

Contribution

It establishes a rigidity result showing that sphere-foliated minimal and constant mean curvature hypersurfaces in hyperbolic product spaces are necessarily rotationally symmetric.

Findings

Hypersurfaces are rotationally symmetric under the given conditions.

Foliation by spheres implies symmetry in hyperbolic product spaces.

Results hold for dimensions n ≥ 3 for minimal and n ≥ 2 for constant mean curvature hypersurfaces.

Abstract

In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero constant mean curvature hypersurfaces when $n\geq2$ foliated by spheres in parallel horizontal hyperplanes in ${\mathbb{H}}^n \times \mathbb{R}$ must be rotationally symmetric.