Year: 2013
Type: preprint
Abstract: A submanifold $M$ of a Euclidean $m$-space is said to be biharmonic if $Δ\overrightarrow H=0$ holds identically, where $\overrightarrow H$ is the mean curvature vector field and $Δ$ 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.
Source: arXiv (Cornell University)
Author: Chen, Bang-Yen
Institution: Michigan State University
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1510027072