发表机构
Institut de Math \'e matiques de Jussieu-Paris Rive Gauche, CNRS UMR 7586, Universit \'e Paris Cit \'e , B \ a timent SoFe Germain Case 7012, 75205 Paris Cedex 13, France.
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明临界悬链面在更弱假设（浸入、可能带分支、嵌入边界）下仍唯一，并应用于$\Bbb S^2$上超定特征值问题，得出解区域为赤道对称的旋转环面。
AI 中文摘要
最近已证明，欧几里得空间单位球$\Bbb B^3$中具有自由边界的嵌入极小环面，在旋转意义下必为临界悬链面。我们证明，在$\Bbb B^3$中具有嵌入边界分支的浸入（可能带分支点）自由边界极小环面必然无分支点且为嵌入。这表明临界悬链面的唯一性在更弱的假设下仍然成立。随后，我们应用这些结果证明：若$\Omega$是$\Bbb S^2$中的一个环面，且存在光滑函数满足超定特征值问题\begin{equation*} \begin{cases} \Delta u+ 2u = 0 \quad \text{on} \quad \Omega\\\\ \qquad \quad u=0 \quad \text{in}\quad\partial\Omega\\\\ \quad\\,\\,\\,\\, |\nabla u|=1 \quad \text{in}\quad \partial\Omega. \end{cases} \end{equation*}则$\Omega$在旋转意义下为具有赤道对称性的旋转环面。
英文摘要
It was recently proved that an embedded minimal annulus with free boundary in the unit ball $\Bbb B^3$ of the Euclidean space is, up to a rotation, the critical catenoid. We prove that an immersed, possibly branched, free boundary minimal annulus in $\Bbb B^3$ with embedded boundary components is necessarily free of branch points and is embedded. This shows the uniqueness of the critical catenoid holds under these weaker hypotheses. We then apply these results to prove that if $Ω$ is an annulus in $\Bbb S^2$ for which there exists a smooth function satisfying the overdetermined eigenvalue problem \begin{equation*} \begin{cases} Δu+ 2u = 0 \quad \text{on} \quad Ω\\ \qquad \quad u=0 \quad \text{in}\quad\partialΩ\\ \quad\,\,\,\, |\nabla u|=1 \quad \text{in}\quad \partialΩ. \end{cases} \end{equation*} then $Ω$ is, up to rotation, a rotational annulus with equatorial symmetry.
