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 $R_*>\pi/2$ and is not injective. Hence each $B(\rho)$ with $\pi/2<\rho \pi/2$. The exact identity $\dim K_0^{ev}(\Sigma_a)=\mathbf{1}_{\{R'=0\}}(a)$ detects the degeneration; it follows from the symmetry-free relation $\partial_\eta\varphi_a-\operatorname{ct}_\kappa(r(a))\varphi_a=r'(a)A_a(\eta,\eta)$, 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.