On biharmonic conformal hypersurfaces

TL;DR

This paper derives biharmonic equations for conformal hypersurfaces in Riemannian manifolds, characterizes when totally umbilical hypersurfaces admit biharmonic conformal immersions, and provides examples in space forms.

Contribution

It generalizes biharmonic equations for conformal hypersurfaces and characterizes conditions for biharmonic conformal immersions of totally umbilical hypersurfaces.

Findings

Conformal factor must be isoparametric for biharmonic conformal immersions of totally umbilical hypersurfaces.

No biharmonic conformal immersion exists for non-minimal totally umbilical hypersurfaces in nonpositive curvature space forms.

A specific hypersphere in a positively curved space form admits a biharmonic conformal immersion.

Abstract

In this paper, we first derive biharmonic equation for conformal hypersurfaces in a generic Riemannian manifold generalizing that for biharmonic hypersurfaces in \cite{Ou1} and that for biharmonic conformal surfaces in \cite{Ou3, Ou2, Ou4}. We then show that if a totally umbilical hypersurface in a space form admits a biharmonic conformal immersion into the ambient space, then the conformal factor has to be an isoparametric function. We also prove that no part of a non-minimal totally umbilical hypersurface in a space form of nonpositive curvature admits a biharmonic conformally immersion into that space form whilst, for the positive curvature space form, we show that the totally umbilical hypersurface $S^4(\frac{\sqrt{3}}{2})\hookrightarrow S^5$ does admit a biharmonic conformal immersion into $S^5$.