Biharmonic hypersurfaces in hemispheres

TL;DR

This paper proves that under certain conditions, the only compact non-minimal biharmonic hypersurfaces in hemispheres are specific small hyperspheres, confirming a conjecture in this geometric setting.

Contribution

It confirms the Balmuș-Montaldo-Oniciuc's conjecture for hemispheres, identifying the unique biharmonic hypersphere under sign conditions on mean curvature.

Findings

The small hypersphere $S^{n}(1/\sqrt{2})$ is the only non-minimal biharmonic hypersurface in a hemisphere under the given conditions.

Biharmonic hypersurfaces in hemispheres are characterized by the sign of $n^2 - H^2$.

The result extends understanding of biharmonic submanifolds in spherical geometries.

Abstract

In this paper we consider the Balmu\c{s}-Montaldo-Oniciuc's conjecture in the case of hemispheres. We prove that a compact non-minimal biharmonic hypersurface in a hemisphere of $S^{n+1}$ must be the small hypersphere $S^{n}\left(1/\sqrt{2}\right)$, provided that $n^{2}-H^{2}$ does not change sign.