Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Bang-Yen Chen

doi:10.3390/math7080710

Abstract

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

Highlights

Let M be a Riemannian submanifold of a Euclidean m-space Em
The study of biharmonic submanifolds was initiated around the middle of 1980s in the author’s program of understanding finite type submanifolds in Euclidean spaces, independently by Jiang in [2] in his study of Euler–Lagrange’s equation of bienergy functional
Dimitric proved in [3,4] that biharmonic curves in Euclidean spaces Em are parts of straight lines; biharmonic submanifolds of finite type in Em are minimal; pseudo umbilical submanifolds M that Em with dim M 6= 4 are minimal, as well as biharmonic hypersurfaces in Em with at most two distinct principal curvatures are minimal

Summary

Introduction

Let M be a Riemannian submanifold of a Euclidean m-space Em. Denote by ∆ the Laplacian of M. The author and Jiang proved independently that there are no biharmonic surfaces in E3 except the minimal ones This non-existence result was later generalized by Dimitric in his doctoral thesis [3] at Michigan State University and his paper [4]. It was known that there exist no biharmonic submanifolds of Em which lie in a hypersphere of Em (see [1], page 181 or [5], Corollary 7.2) Based on these results mentioned above, the author made in 1991 the following conjecture (see, e.g., [1,6,7]): Chen’s Conjecture. The study of submanifolds with parallel normalized mean curvature vector in Euclidean spaces was initiated at the beginning of the 1980s (see [32]). A biharmonic surface in Em with a parallel normalized mean curvature vector does not exist

Preliminaries

Proof of the Theorem 1

Some Remarks