Complete cohomogeneity one hypersurfaces of $\mathbb{H}^{n+1}$

TL;DR

This paper classifies certain complete hypersurfaces in hyperbolic space that admit a symmetry group acting with cohomogeneity one, providing geometric characterizations based on the manifold's compactness and curvature properties.

Contribution

It offers a new characterization of cohomogeneity one hypersurfaces in hyperbolic space under specific conditions on the manifold's dimension and curvature.

Findings

Characterization of isometric immersions with symmetry in hyperbolic space.

Conditions for compactness and curvature influence on hypersurface classification.

Extension of known results to higher dimensions and specific curvature sets.

Abstract

We study isometric immersions $f: M^n \rightarrow \mathbb{H}^{n+1}$ into hyperbolic space of dimension $n+1$ of a complete Riemannian manifold of dimension $n$ on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We provide a characterization if either $n \geq 3$ and $M^n$ is compact, or $n \geq 5$ and the connected components of the set where the sectional curvature is constant and equal to $-1$ are bounded.