A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere
Abstract
Let {Σa}, a∈(0,1/2), be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls B(R(a))⊂𝕊3, R(a)>π/2. We prove that R is real-analytic, tends to π/2 at both ends, and therefore folds: it has an interior maximum R*>π/2 and is not injective. Hence each B(ρ) with π/2<ρ<R* contains at least two, and only finitely many, mutually non-congruent annuli of the family. At every critical point of R the annulus is degenerate modulo ball-preserving isometries: its Jacobi--Robin kernel contains a rotationally invariant, reflection-even field not induced by any Killing field of 𝕊3 preserving the ball; its nullity is at least three. These degenerate annuli form a nonempty discrete set; off it, the rotationally invariant even nullity vanishes. Those sitting at a maximizer of R, in the largest cap B(R*), are called degenerate annuli of maximal cap radius; each such annulus is also a strict local area maximizer in the family, since area and R have the same critical points. Thus the hypothesis that all Jacobi fields of an embedded free boundary minimal annulus in a spherical cap are Killing-induced, used in the Naff--Zhu continuity approach to uniqueness, fails for some radius R>π/2. The exact identity dim K0ev(Σa)=1{R'=0}(a) detects the degeneration; it follows from the symmetry-free relation ∂ηφa−ctκ(r(a))φa=r'(a)Aa(η,η), a Robin defect identity, which requires of the ambient only that the barrier family be umbilic and which extends to capillary boundary conditions at constant contact angle.
Related
- Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
- Uniqueness of free boundary minimal annuli
- The Frohmanian Symplectic Tether: Canonical Uniqueness on Coadjoint Orbits of SDiff(T3), Non-Degenerate Casimir Coupling, and Conditional Global Regularity via Quartic Young Absorption in Lean 4
- Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
- Grushin spherical harmonics in non-degenerate dimension one
BibTeX
@misc{pigazzini2026degenerate,
title = {{A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere}},
author = {Alexander Pigazzini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.23534},
eprint = {2607.23534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23534},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere
AU - Alexander Pigazzini
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2607.23534
UR - https://arxiv.org/abs/2607.23534
AB - Let {Σ_a}, a∈(0,1/2), be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls B(R(a))⊂𝕊^3, R(a)>π/2. We prove that R is real-analytic, tends to π/2 at both ends, and therefore folds: it has an interior maximum R_*>π/2 and is not injective. Hence each B(ρ) with π/2<ρ<R_* contains at least two, and only finitely many, mutually non-congruent annuli of the family. At every critical point of R the annulus is degenerate modulo ball-preserving isometries: its Jacobi--Robin kernel contains a rotationally invariant, reflection-even field not induced by any Killing field of 𝕊^3 preserving the ball; its nullity is at least three. These degenerate annuli form a nonempty discrete set; off it, the rotationally invariant even nullity vanishes. Those sitting at a maximizer of R, in the largest cap B(R_*), are called degenerate annuli of maximal cap radius; each such annulus is also a strict local area maximizer in the family, since area and R have the same critical points. Thus the hypothesis that all Jacobi fields of an embedded free boundary minimal annulus in a spherical cap are Killing-induced, used in the Naff--Zhu continuity approach to uniqueness, fails for some radius R>π/2. The exact identity dim K_0^ev(Σ_a)=1_{R'=0}(a) detects the degeneration; it follows from the symmetry-free relation ∂_ηφ_a−ct_κ(r(a))φ_a=r'(a)A_a(η,η), a Robin defect identity, which requires of the ambient only that the barrier family be umbilic and which extends to capillary boundary conditions at constant contact angle.
ER -