preprint
Inserted: 29 jul 2026
Last Updated: 4 aug 2026
Year: 2026
Abstract:
Let $\{\Sigma_a\}_{a\in(0,1/2)}$ be de Oliveira's family of embedded free boundary minimal annuli of revolution contained in geodesic balls $B(R(a))\subset\mathbb{S}^3$ of radius $R(a)>\tfrac\pi2$. We prove that the radius map $R$ is real-analytic and tends to $\tfrac\pi2$ at both ends of the parameter range, and therefore folds: it is non-monotone, attains an interior maximum $R_*>\tfrac\pi2$, and is not injective. Two consequences follow. First, every geodesic ball of radius strictly between $\tfrac\pi2$ and $R_*$ contains at least two, and at most finitely many, mutually non-congruent embedded free boundary minimal annuli of the family. Second, at every critical point of $R$ the annulus is degenerate: it carries a rotationally invariant, reflection-even Jacobi--Robin field which is not induced by any Killing field of $\mathbb{S}^3$ preserving the ball, and its nullity is at least three. The critical set is a nonempty discrete subset of $(0,\tfrac12)$, so these annuli are isolated in a family whose remaining members have vanishing nullity in that sector; those sitting at a maximizer of $R$, hence in the largest cap $B(R_*)$ the family reaches, we call degenerate annuli of maximal cap radius. Along the family the critical points of the area coincide with those of $R$, so that a degenerate annulus of maximal cap radius is also a local maximizer of the area within the family. In particular the hypothesis that all Jacobi fields of an embedded free boundary minimal annulus in a spherical cap are Killing-induced, which underlies the continuity approach to uniqueness of Naff--Zhu, does not survive the passage beyond $R=\tfrac\pi2$. The degeneration is detected by the exact identity $\dim Ker_0^{\mathrm{ev}}(\Sigma_a)=\mathbf{1}_{\{r'=0\}}(a)$, itself a consequence of a boundary relation valid in every space form and every dimension, with no symmetry assumption: for a family of free boundary minimal hypersurfaces in concentric geodesic balls $B(r(a))$, the normal variation field satisfies a Robin defect identity: $\partial_\eta\varphi_a-\mathrm{ct}(r(a))\varphi_a=r'(a)A_a(\eta,\eta)$.
Keywords: Free boundary minimal surfaces, spherical caps, minimal annuli, degenerate annulus, non-uniqueness, Jacobi fields, Robin boundary condition, nullity
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