On the normal stability of the 4-harmonic and the ES-4-harmonic hypersphere
Volker Branding
TL;DR
The paper proves that the small hypersphere $ι: \mathbb{S}^m(\tfrac{1}{2})\to\mathbb{S}^{m+1}$ is unstable under normal variations for both the 4-energy $E_4$ and the ES-4-energy $E^{ES}_4$, with the normal index equal to 1 in each case. It derives a comprehensive second variation framework for these higher-order energies, first in general, then in space-forms, and finally for normal variations, enabling explicit stability tests via the Laplace–Beltrami spectrum. For the small hypersphere, explicit data $A=-\sqrt{3}I$, $H=-\sqrt{3}$, and $|A|^2=3m$ lead to a detailed quadratic form $Q_4(f\nu,f\nu)$, whose analysis together with $\lambda_1=4m$ confirms $\operatorname{Ind}^{\mathrm{nor}}_4=1$; a parallel computation for the ES-4 curvature term yields $\operatorname{Ind}^{\mathrm{nor}}_{ES-4}=1$. The results support a broader conjecture that normal indices of small $k$-harmonic hyperspheres in space forms are typically 1 and show that harmonic maps are weakly stable for the ES-4 curvature functional, highlighting robust aspects of normal stability across higher-order generalizations.
Abstract
Both 4-harmonic and ES-4-harmonic maps are two higher order generalizations of the well-studied harmonic map equation given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. It is well known that the small hypersphere \(ι\colon\s^m(\frac{1}{2})\to\s^{m+1}\) is a critical point of both the 4-energy as well as the ES-4-energy but up to now it has not been investigated if there is a difference concerning its stability. The main contribution of this article is to show that the small hypersphere is unstable with respect to normal variations both as 4-harmonic hypersphere as well as ES-4-harmonic hypersphere and that its normal index equals one in both cases.