Minimal hypersurfaces in R^n \times S^m

TL;DR

This paper classifies certain symmetric minimal hypersurfaces in product spaces and constructs examples of constant mean curvature hypersurfaces, analyzing their stability properties.

Contribution

It provides a classification of invariant minimal hypersurfaces in $R^n imes S^m$ and constructs new constant mean curvature examples, exploring their stability.

Findings

Classified invariant minimal hypersurfaces in $R^n imes S^m$.

Constructed compact and noncompact constant mean curvature hypersurfaces.

Showed all such hypersurfaces are unstable.

Abstract

We classify minimal hypersurfaces in $R^n \times S^m$, $n,m \geq 2$, which are invariant by the canonical action of $O(n) \times O(m)$. We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature. We show that the minimal hypersurfaces and the noncompact constant mean curvature hypersurfaces are all unstable.