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 Bn, 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 ind(TC)=n, 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 C2 free boundary minimal deformations.
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BibTeX
@misc{matinpour2026rigidity,
title = {{Rigidity and Morse index of free boundary minimal $T$-surfaces in the unit ball}},
author = {Elham Matinpour},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.27776},
eprint = {2605.27776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.27776},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rigidity and Morse index of free boundary minimal T-surfaces in the unit ball
AU - Elham Matinpour
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2605.27776
UR - https://arxiv.org/abs/2605.27776
AB - 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 ind(TC)=n, 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.
ER -