On Steklov eigenspaces for free boundary minimal surfaces in the unit ball (Q6619989)

The paper under review develops some new methods to compare the space \(\mathcal{C}(\Sigma)\) spanned by the coordinate functions on a free boundary minimal submanifolds \(\Sigma\) embedded in the Euclidean unit \(n\)-ball \(\mathbb{B}^n\) with its first Steklov eigenspace \(\mathcal{E}_{\sigma_1}(\Sigma)\).\N\NFirst, the authors show that \(\mathcal{C}(A)=\mathcal{E}_{\sigma_1}(A)\) holds for any embedded antipodally-invariant free boundary minimal annulus \(A\) in \(\mathbb{B}^3\), which combined with a result of \textit{A. Fraser} and \textit{R. Schoen} [Invent. Math. 203, No. 3, 823--890 (2016; Zbl 1337.35099)] implies that \(A\) is congruent to the critical catenoid. This generalises the previous result by the second author [Indiana Univ. Math. J. 67, No. 2, 889--897 (2018; Zbl 1455.53032)] and confirms a conjecture of \textit{A. Fraser} and \textit{M. M. C. Li} [J. Differ. Geom. 96, No. 2, 183--200 (2014; Zbl 1295.53062)] under the additional antipodally-invariant assumption.\N\NThen the authors show that \(\mathcal{C}(\Sigma)=\mathcal{E}_{\sigma_1}(\Sigma)\) for any embedded free boundary minimal surface \(\Sigma\) in \(\mathbb{B}^3\) under the assumption that \(\Sigma\) is invariant under a group of isometries \(G\) satisfying certain technical properties. This result applies to many known examples or expected examples of free boundary minimal surfaces in \(\mathbb{B}^3\). The authors describe these applications in detail in the last section of this paper. Meanwhile, the authors prove an analogous result in the setting of embedded minimal surfaces in \(\mathbb{S}^3\).\N\NIn general, it is conjectured in this paper that \(\mathcal{C}(\Sigma)=\mathcal{E}_{\sigma_1}(\Sigma)\) holds for any properly embedded free boundary minimal hypersurface \(\Sigma\subset \mathbb{B}^n\). This is a strengthening of the Fraser-Li conjecture: the first Steklov eigenvalue \(\sigma_1\) of a properly embedded free boundary minimal hypersurface \(\Sigma\) in \(\mathbb{B}^n\) is equal to \(1\).