Normal index of the small k-harmonic hypersphere

Establish that the normal index of the small k-harmonic hypersphere φ: S^m(1/√k) → S^{m+1} equals one for all k, where the normal index denotes the maximal dimension of any linear subspace of normal variations on which the second variation of the k-energy is negative.

Background

The paper studies the normal stability of the small hypersphere S^m(1/2) in S^{m+1} as a critical point of both the 4-energy (for 4-harmonic maps) and the ES-4-energy (for ES-4-harmonic maps). It proves that the normal index equals one in both settings, thereby providing evidence related to broader stability patterns for higher-order harmonic hypersurfaces.

Within this broader context, the authors explicitly reference a conjecture from the literature asserting that the normal index of the small k-harmonic hypersphere S^m(1/√k) in S^{m+1} is always one. Their new results for k=4 and ES-4 lend further support, but the general case across all k remains conjectural.