Four dimensional biharmonic hypersurfaces in nonzero space form have constant mean curvature

TL;DR

This paper proves that four-dimensional biharmonic hypersurfaces in nonzero space forms necessarily have constant mean curvature, confirming a conjecture for this specific case.

Contribution

It establishes that such hypersurfaces must have constant mean curvature, providing a positive resolution to a 2008 conjecture for four-dimensional cases.

Findings

Four-dimensional biharmonic hypersurfaces in nonzero space forms have constant mean curvature.

The result confirms the Balmus-Montaldo-Oniciuc conjecture for four-dimensional hypersurfaces.

Analysis of Gauss and Codazzi equations was key to the proof.

Abstract

In this paper, through making careful analysis of Gauss and Codazzi equations, we prove that four dimensional biharmonic hypersurfaces in nonzero space form have constant mean curvature. Our result gives the positive answer to the conjecture proposed by Balmus-Montaldo-Oniciuc in 2008 for four dimensional hypersurfaces.