Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal

TL;DR

This paper proves that weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are necessarily minimal, confirming a special case of Chen's conjecture for these geometric conditions.

Contribution

It establishes that weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal, extending previous results to a broader class of ambient spaces.

Findings

Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal.

Weakly convex hypersurfaces with harmonic mean curvature vector fields in Euclidean spaces are minimal.

Supports Chen's conjecture in specific geometric contexts.

Abstract

A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $\Delta \vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $\Delta$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositive curved space forms are minimal.