A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere

Abstract

Let $\{{\Sigma }_a\}$, $a\in (0,1/2)$, be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls $B(R(a))\subset {\mathbb{S}}^3$, $R(a)>\pi /2$. We prove that $R$ is real-analytic, tends to $\pi /2$ at both ends, and therefore folds: it has an interior maximum $R_{\ast }>\pi /2$ and is not injective. Hence each $B(\rho )$ with $\pi /2<\rho <R_{\ast }$ 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 ${\mathbb{S}}^3$ 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 $B(R_{\ast })$, 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 $R>\pi /2$. The exact identity $\dim K_0^{ev}({\Sigma }_a)=1_{\{R^{\prime }=0\}}(a)$ detects the degeneration; it follows from the symmetry-free relation ${\partial }_{\eta }{\phi }_a-{\mathrm{ct}}_{\kappa }(r(a)){\phi }_a=r^{\prime }(a)A_a(\eta ,\eta )$, 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.

v5: corrected attribution of Problem 6.2 to [20, Thesis, Rem. 4.3]; added comparison with the spectral characterisations of [20, Thesis, Prop. 2.6, Thms. 2.7-2.8]; corrected attribution of the first part of Lemma 2.5, it is the minimal case of Corollary 4.4 in [6]. No change to the results