Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$f$-Biharmonic hypersurfaces into a conformally flat space

Ze-Ping Wang, Li-Hua Qin, Xue-Yi Chen

Abstract

We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We also explore $f$-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Secondly, we construct $f$-biharmonic surfaces and biharmonic conformal immersions of the associated surfaces into a conformall flat 3-space and also give a complete classification of $f$-biharmonic surfaces of nonzero constant mean curvature in 3-space forms. Finally, we especially investigate $f$-biharmonicity of hypersurfaces into a conformally flat space of negative sectional curvature. We show that any totally umbilical $f$-biharmonic surface of a 3-manifold with nonpositve sectional curvature is minimal whilst there are proper $f$-biharmonic $m$-dimensional submanifolds with $m\geq3$ and $m\neq4$ into nonpositvely curved manifolds.

$f$-Biharmonic hypersurfaces into a conformally flat space

Abstract

We first study -biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper -biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We also explore -biharmonicity of totally umbilical hyperplanes in a conformally flat space. Secondly, we construct -biharmonic surfaces and biharmonic conformal immersions of the associated surfaces into a conformall flat 3-space and also give a complete classification of -biharmonic surfaces of nonzero constant mean curvature in 3-space forms. Finally, we especially investigate -biharmonicity of hypersurfaces into a conformally flat space of negative sectional curvature. We show that any totally umbilical -biharmonic surface of a 3-manifold with nonpositve sectional curvature is minimal whilst there are proper -biharmonic -dimensional submanifolds with and into nonpositvely curved manifolds.

Paper Structure

This paper contains 4 sections, 23 theorems, 52 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Totally umbilical $f$-biharmonic hypersurfaces
  3. $f$-Biharmonic surfaces and biharmonic conformal immersions
  4. Proper $f$-biharmonic hypersurfaces $M^m$ with $m\geq3$

Key Result

Theorem 1.1

$($Ou5Ou3$)$ A hypersurface $\phi:(M^{m},g)\to (N^{m+1},h)$ with mean curvature vector field $\eta=H\xi$ is $f$-biharmonic if and only if: which is equivalent to

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • proof
  • Remark 1
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 30 more