The Morse index of prismatic free boundary minimal surfaces in the unit ball

@misc{tran2026morse,
  title = {{The Morse index of prismatic free boundary minimal surfaces in the unit ball}},
  author = {Hung Tran},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02078},
  eprint = {2610.02078},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02078},
}

TY  - UNPB
TI  - The Morse index of prismatic free boundary minimal surfaces in the unit ball
AU  - Hung Tran
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02078
UR  - https://arxiv.org/abs/2610.02078
AB  - For integer b≥3, let Sig⊂Ball be an embedded free boundary minimal surface (FBMS) of genus zero with b boundary components. Suppose that Sig is invariant under the prismatic group of order 4b, the rotation of order b permutes the boundary components cyclically, and the reflection in the equatorial plane preserves each of them. We show that the Morse index of Sig is 2b and its nullity is three; moreover, we determine the negative space of the index form as a representation of the symmetry group. Examples include the b-noids of Karpukhin, Kusner, McGrath, and Stern and, for large b, the genus-zero surfaces of Folha, Pacard, and Zolotareva. To our knowledge, apart from the equatorial disc and the critical catenoid, these are the first FBMS in the ball whose Morse indices are known precisely. There are also lower bounds for the index of surfaces with rotational or polyhedral symmetry. The main new ingredient is an equivariant comparison, using the representation theory of finite groups, between area and energy, in which the Teichmüller directions are counted by symmetry type using circle domains.
ER  -