Delaunay-type hypersurfaces in cohomogeneity one manifolds

TL;DR

This paper proves the existence of Delaunay-type hypersurfaces with constant mean curvature in various compact manifolds, extending classical examples to new ambient spaces using symmetry and bifurcation methods.

Contribution

It introduces a new construction of Delaunay-type hypersurfaces in diverse compact manifolds via cohomogeneity one group actions and variational bifurcation techniques.

Findings

Existence of Delaunay-type hypersurfaces in complex and quaternionic projective spaces

Construction of hypersurfaces in Kervaire exotic spheres

Extension of classical Delaunay surfaces to broader geometric contexts

Abstract

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction specializes to the classical examples in round spheres, and allows to obtain Delaunay-type hypersurfaces in many other ambient spaces, ranging from complex and quaternionic projective spaces to Kervaire exotic spheres.