Free-boundary analogue of the ambient-instability lower bound

Establish an analogue for free boundary minimal surfaces in the Euclidean unit ball $\mathbb{B}^{n+1}$ of Proposition 2.7, which states that for a conformal harmonic immersion into $\mathbb{S}^n$ not contained in a totally geodesic subsphere, the ambient representation embeds into the representation on the negative eigenspace of the energy index form.

Background

For minimal surfaces in the round sphere, Proposition 2.7 supplies a representation-theoretic lower bound for the energy index by constructing an $(n+1)$-dimensional space of negative directions from projections of constant ambient vectors. This lower bound combines with the spectral index-one condition to determine the full energy representation.

The authors explicitly state that they do not know a corresponding result for free boundary minimal surfaces in the unit ball. In the paper, the energy indices of the free boundary examples are instead obtained indirectly from area-index estimates and symmetry decompositions.