Complete Hypersurfaces of Constant Isotropic Curvature in Space Forms

TL;DR

This paper classifies complete hypersurfaces with constant isotropic curvature in space forms, revealing conditions under which they have constant mean curvature or are minimal, and identifying specific geometric structures involved.

Contribution

It provides a complete classification of such hypersurfaces, establishing their mean curvature properties and characterizing minimal cases explicitly.

Findings

Hypersurfaces with constant isotropic curvature have constant mean curvature iff they are isoparametric.

Minimal hypersurfaces are either totally geodesic or Clifford minimal hypersurfaces.

Explicit identification of Clifford minimal hypersurface in the sphere.

Abstract

We classify complete orientable hypersurfaces of constant isotropic curvature in space forms. We show that such a hypersurface has constant mean curvature only if it is an isoparametric hypersurface, and that it is minimal if and only if it is totally geodesic or it is the Clifford minimal hypersurface ${\mathbb S}^{3}(\frac{4c}{3})\times {\mathbb S}^{1}(4c)$ in ${\mathbb S}^{5}(c).$