2021/04/19 by Hang Chen, Chen, Hang, Zhida Guan +1
Mathematics · #53C43 (Primary) 53C42 (Secondary) #58E20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2104.09287
openalex publication_date 2021/04/19 · openalex created_date 2022/12/14 · openalex updated_date 2026/07/28
A k-harmonic map is a critical point of the k-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if Mn (n≥ 3) is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form ℝn+1(c), then M has constant scalar curvature. This supports the generalized Chen's conjecture when c≤ 0. When c=1, we give an optimal upper bound of the mean curvature H for a non-totally umbilical proper CMC k-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we give the complete classification of the 3-dimensional closed proper CMC triharmonic hypersurfaces in \mathbbS4.