On the normal stability of triharmonic hypersurfaces in space forms
Volker Branding
TL;DR
This work extends higher order variational analysis to triharmonic maps and hypersurfaces, focusing on normal stability in space forms. It derives the full first and second variation formulas for the trienergy E_3 and specializes to space forms to obtain tractable Hessians. The authors prove that triharmonic hypersurfaces with constant mean curvature are weakly normally stable in Euclidean space and normally stable in hyperbolic space, and they determine the normal index for the small proper triharmonic sphere to be one, with further analysis and bounds for the generalized Clifford torus. These results deepen the understanding of stability for higher order variational problems and provide concrete criteria for normal stability and normal index in classical space forms.
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 $φ\colon\mathbb{S}^m(1/\sqrt{3})\hookrightarrow\mathbb{S}^{m+1}$ is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.