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On biharmonic conformal hypersurfaces

A. Mohammed Cherif, Ye-Lin Ou

TL;DR

The paper derives a general biharmonic equation for conformal hypersurfaces $\tau_2(\phi)=0$ in a Riemannian ambient and shows that, for non-minimal totally umbilical hypersurfaces in space forms, the conformal factor must be isoparametric with explicit expressions for $|\nabla \lambda|^2$ and $\Delta \lambda$. It then establishes strong nonexistence results in nonpositive curvature for $m\ge 4$ and demonstrates existence of nontrivial examples in positive curvature, notably via biharmonic conformal immersions of $S^4(a) \hookrightarrow S^5$ with a conformal factor $\lambda=(1+|x|^2)^{-1}$; in particular, the case $m=4$ yields concrete parameter values $a=1$ or $a=\sqrt{3}/2$ leading to explicit constructions. The work unifies conformal and submanifold biharmonic theories, provides explicit ODEs and isoparametric constraints for the conformal factor, and offers concrete geometric examples in Euclidean, hyperbolic, and spherical settings.

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 $S^5$.

On biharmonic conformal hypersurfaces

TL;DR

The paper derives a general biharmonic equation for conformal hypersurfaces in a Riemannian ambient and shows that, for non-minimal totally umbilical hypersurfaces in space forms, the conformal factor must be isoparametric with explicit expressions for and . It then establishes strong nonexistence results in nonpositive curvature for and demonstrates existence of nontrivial examples in positive curvature, notably via biharmonic conformal immersions of with a conformal factor ; in particular, the case yields concrete parameter values or leading to explicit constructions. The work unifies conformal and submanifold biharmonic theories, provides explicit ODEs and isoparametric constraints for the conformal factor, and offers concrete geometric examples in Euclidean, hyperbolic, and spherical settings.

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 does admit a biharmonic conformal immersion into .
Paper Structure (2 sections, 8 theorems, 75 equations)

This paper contains 2 sections, 8 theorems, 75 equations.

Table of Contents

  1. Biharmonic conformal hypersurfaces: Equations and some examples
  2. Biharmonic conformal immersions of non-minimal totally umbilical hypersurfaces

Key Result

Theorem 1.1

A conformal immersion $\phi : (M^m, \overline{g})\to (N^{m+1}, h)$ with $\phi^*h=\lambda^2\bar{g}$ is biharmonic if and only if where $\xi$, $A$, and $H$ are the unit normal vector field, the shape operator, and the mean curvature function of the hypersurface $\phi(M)\subset (N^{m+1}, h)$ respectively, and the operators $\Delta,\; \nabla$ and $|,|$ are taken with respect to the induced metric $g=

Theorems & Definitions (20)

  • Theorem 1.1
  • proof
  • Remark 1
  • Corollary 1.2
  • proof
  • Corollary 1.3
  • proof
  • Example 1
  • Example 2
  • Example 3
  • ...and 10 more