Recent progress of biharmonic hypersurfaces in space forms

Abstract

In the past two decades, biharmonic submanifolds have attracted much attention from mathematicians all over the world. In particular, concerning Chen’s conjecture and the generalized Chen’s conjecture, many meaningful results have been obtained. In this survey paper, we will restrict our attention to biharmonic hypersurfaces in the real space forms. In the first three parts, we give a short survey on some new developments on biharmonic hypersurfaces in Euclidean spaces, hyperbolic spaces and Euclidean spheres, respectively. In the last section, we outline the proof of Chen’s conjecture for hypersurfaces of R 5 \mathbb {R}^5 [Adv. Math, 383 (2021), Paper No. 107697, 28] and point out the potential importance of the method and the key techniques used in the proof for further studies on biharmonic hypersurfaces in space forms.