Triharmonic CMC hypersurfaces in space forms with 4 distinct principal curvatures

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 ≥ 4) is a CMC proper triharmonic hypersurface in a space form 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 R 5 (c) is minimal when c ≤ 0, which supports the generalized Chen's conjecture. We also give some characterizations of CMC proper triharmonic hypersurfaces in S 5 .