Triharmonic CMC hypersurfaces in space forms with at most 3 distinct principal curvatures

TL;DR

This paper classifies certain special hypersurfaces with constant mean curvature and limited principal curvatures in space forms, proving they have constant scalar curvature and providing a complete classification in the 3D sphere case.

Contribution

It proves that proper triharmonic hypersurfaces with at most three principal curvatures in space forms have constant scalar curvature, supporting Chen's conjecture for non-positive curvature, and classifies 3D cases in spheres.

Findings

Proper triharmonic hypersurfaces with limited principal curvatures have constant scalar curvature.

Supports Chen's conjecture when the ambient space has non-positive curvature.

Complete classification of 3D proper CMC triharmonic hypersurfaces in spheres.

Abstract

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 $M^{n} (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form $\mathbb{R}^{n+1}(c)$, then $M$ has constant scalar curvature. This supports the generalized Chen's conjecture when $c\le 0$. 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 $\mathbb{S}^{4}$.