Chen's biharmonic conjecture is well-known and stays open: The only

biharmonic submanifolds of Euclidean spaces are the minimal ones. In

this paper, we consider an advanced version of the conjecture,

replacing $Delta$ by its extension, $L_1$-operator

($L_1$-conjecture). The $L_1$-conjecture states that any

$L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that

the $L_1$-conjecture is true for $L_1$-biharmonic hypersurfaces with

three distinct principal curvatures and constant mean curvature of a

Euclidean space of arbitrary dimension.

Type of Study: Research paper |
Subject:
Special