Recent developments of biharmonic conjecture and modified biharmonic conjectures

TL;DR

This paper surveys recent progress on the biharmonic conjecture and its variants, which relate to the minimality of biharmonic submanifolds in Euclidean spaces, highlighting new developments and proposing modified conjectures.

Contribution

The paper reviews recent advances in biharmonic submanifold research and introduces two new modified conjectures related to biharmonicity.

Findings

Progress on the original biharmonic conjecture since 2000

Development of generalized biharmonic conjectures

Introduction of two modified 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.