AI 中文总结
研究球冠中自由边界极小旋转环面族，证明相关半径函数\(R\)的性质及环面退化情况，指出纳夫 - 朱唯一性假设对某些\(R>\pi/2\)不成立，通过罗宾缺陷等式检测退化，给出环面维度等式。
AI 中文摘要
设\(\{\Sigma_a\}\)，\(a\in(0,1/2)\)，是德奥利维拉在测地球\(B(R(a))\subset\mathbb{S}^3\)（\(R(a)>\pi/2\)）中嵌入的自由边界极小旋转环面族。我们证明\(R\)是实解析的，在两端趋于\(\pi/2\)，因此有折叠：它有一个内部最大值\(R_*>\pi/2\)且不是单射的。所以每个\(\pi/2<\rho<R_*\)的\(B(\rho)\)至少包含两个且只有有限多个该族中相互不全等的环面。在\(R\)的每个临界点，环面关于保球等距变换是退化的：其雅可比 - 罗宾核包含一个旋转不变、反射偶的场，不由\(\mathbb{S}^3\)中任何保球的克利福德场诱导；其零度至少为三。这些退化环面形成一个非空离散集；在其之外，旋转不变偶零度消失。那些位于\(R\)的最大值点、在最大球冠\(B(R_*)\)中的环面称为最大球冠半径的退化环面；每个这样的环面在族中也是严格局部面积最大化者，因为面积和\(R\)有相同的临界点。因此，在纳夫 - 朱唯一性连续性方法中使用的关于球冠中嵌入自由边界极小环面的所有雅可比场都是克利福德诱导的假设，对于某些\(R>\pi/2\)不成立。精确等式\(\dim K_0^{ev}(\Sigma_a)=\mathbf{1}_{\{R'=0\}}(a)\)检测到退化；它由所有空间形式和维度中的无对称关系\(\partial_\eta\varphi_a-\operatorname{ct}_\kappa(r(a))\varphi_a=r'(a)A_a(\eta,\eta)\)，一个罗宾缺陷等式得出。
英文摘要
Let $\{Σ_a\}$, $a\in(0,1/2)$, be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls $B(R(a))\subset\mathbb{S}^3$, $R(a)>π/2$. 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 $\mathbb{S}^3$ 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 $B(R_*)$, 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 $\dim K_0^{ev}(Σ_a)=\mathbf{1}_{\{R'=0\}}(a)$ detects the degeneration; it follows from the symmetry-free relation $\partial_ηφ_a-\operatorname{ct}_κ(r(a))φ_a=r'(a)A_a(η,η)$, 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.
Commentsv5: 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
