Triharmonic CMC hypersurfaces in space forms with 4 distinct principal curvatures

TL;DR

This paper investigates proper triharmonic hypersurfaces with constant mean curvature in space forms, proving they have constant scalar curvature under certain conditions and supporting the generalized Chen's conjecture.

Contribution

It establishes that such hypersurfaces with four distinct principal curvatures have constant scalar curvature, and confirms minimality in specific cases, advancing understanding of triharmonic submanifolds.

Findings

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

In Euclidean space, these hypersurfaces are minimal when the ambient space has non-positive curvature.

Characterizations of CMC proper triharmonic hypersurfaces in the sphere are provided.

Abstract

A triharmonic map is a critical point of the tri-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^n (n\ge 4)$ is a CMC proper triharmonic hypersurface in a space form $\mathbb{R}^{n+1}(c)$ with four distinct principal curvatures and the multiplicity of the zero principal curvature is at most one, then $M$ has constant scalar curvature. In particular, we obtain any CMC proper triharmonic hypersurface in $\mathbb{R}^5(c)$ is minimal when $c\le 0$, which supports the generalized Chen's conjecture. We also give some characterizations of CMC proper triharmonic hypersurfaces in $\mathbb{S}^5$.