Recent developments of biharmonic conjecture and modified biharmonic conjectures

Abstract

A submanifold $M$ of a Euclidean $m$-space is said to be biharmonic if $\Delta \overrightarrow{H}=0$ holds identically, where $\overrightarrow{H}$ is the mean curvature vector field and $\Delta$ is the Laplacian on $M$. In 1991, the author conjectured that every biharmonic submanifold of a Euclidean space is minimal. The study of biharmonic submanifolds is nowadays a very active research subject. In particular, since 2000 biharmonic submanifolds have been receiving a growing attention and have become a popular subject of study with many progresses. In this article, we provide a brief survey on recent developments concerning my original conjecture and generalized biharmonic conjectures. At the end of this article, I present two modified conjectures related with biharmonic submanifolds.

8 pages, to appear in Proceedings of PADGE-2012 (in honor of Franki Dillen)

References in corpus (3)

• Properties of biharmonic submanifolds in spheres
• Biminimal properly immersed submanifolds in complete Riemannian manifolds of non-positive curvature
• On conformal biharmonic immersions

Cited by in corpus (10)

• Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space
• On cohomogeneity one biharmonic hypersurfaces into the Euclidean space
• Reduction methods for the bienergy
• Some open problems and conjectures on submanifolds of finite type: recent development
• f-Biharmonic and bi-f-harmonic submanifolds of generalized space forms
• $\delta$(3)-ideal null 2-type hypersurfaces in Euclidean spaces
• Polyharmonic maps of order k with finite L^p k-energy into Euclidean spaces
• Triharmonic Riemannian submersions from 3-dimensional manifolds of constant curvature
• Biharmonic Immersion in Cartan Hadamard manifold
• Notes on isotropic geometry of production models