Rigidity and Morse index of free boundary minimal $T$-surfaces in the unit ball

Abstract

We study regular two-dimensional free boundary minimal $T$-surfaces in the unit ball $\B^n$, with six faces, four triple junction arcs, and one tetrahedral vertex. A conformal minimal immersion of the standard flat $T$-cone, with the prescribed matching of junction traces, has the flat cone as its image. We also obtain rigidity criteria that do not require matching conformal parametrizations. In the disk-type class, two opposite planar faces suffice. More generally, it is enough that a natural junction invariant vanish on both junction edges of two opposite faces. For variations preserving the junction identifications and keeping the outer boundary on the sphere, we compute \[ \operatorname{ind}(TC)=n, \qquad \operatorname{nul}(TC)=3(n-2). \] The normal kernel consists exactly of the normal components of ambient rotations, and we compute the associated coupled harmonic Steklov spectrum. Every regular free boundary minimal $T$-surface with six faces has index at least $n$, with equality if and only if it is the flat $T$-cone; the same characterization holds for ambient variations. Finally, the flat cone is locally rigid, modulo orthogonal transformations, among sufficiently small compatible facewise $C^2$ free boundary minimal deformations.