On the normal stability of triharmonic hypersurfaces in space forms

TL;DR

This paper investigates the stability properties of triharmonic hypersurfaces in space forms, revealing conditions under which they are stable or weakly stable, with specific results for Euclidean, hyperbolic, and spherical spaces.

Contribution

It provides new stability results for triharmonic hypersurfaces in various space forms, including Euclidean, hyperbolic, and spherical spaces, with detailed analysis of their normal stability.

Findings

Euclidean space hypersurfaces with constant mean curvature are weakly stable

Hyperbolic space hypersurfaces with constant mean curvature are stable

The normal index of the small proper triharmonic hypersphere is one

Abstract

This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere $\phi\colon\mathbb{S}^m(1/\sqrt{3})\hookrightarrow\mathbb{S}^{m+1}$ is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.