Closed Biconservative Hypersurfaces in Spheres

TL;DR

This paper characterizes non-constant mean curvature biconservative hypersurfaces in spheres as special elastic curves, establishing a family of non-embedded, closed examples in space forms.

Contribution

It introduces a novel link between biconservative hypersurfaces and p-elastic curves, providing a classification and existence results for closed non-CMC examples.

Findings

Existence of a discrete biparametric family of non-CMC closed hypersurfaces.

None of these hypersurfaces can be embedded in the sphere.

Characterization of profile curves as p-elastic curves.

Abstract

We characterise the profile curves of non-CMC biconservative rotational hypersurfaces of space forms $N^n(\rho)$ as $p$-elastic curves, for a suitable rational number $p\in[1/4,1)$ which depends on the dimension $n$ of the ambient space. Analysing the closure conditions of these $p$-elastic curves, we prove the existence of a discrete biparametric family of non-CMC closed (i.e., compact without boundary) biconservative hypersurfaces in $\mathbb{S}^n(\rho)$. None of these hypersurfaces can be embedded in $\mathbb{S}^n(\rho)$.