Analytic saddle spheres in \(\mathbb{S}^{3}\) are equatorial (Q6583562)

The paper under review explores global geometric properties of closed surfaces in the three-dimensional sphere \(\mathbb S^3\). As the main result, the authors prove the following fundamental theorem which extends the Calabi-Almgren theorem on minimal spheres in \(\mathbb S^3\) and can be viewed as a spherical version of the Bernstein theorems on minimal and saddle entire graphs in the Euclidean space \(\mathbb E^3\), see [\textit{F. Almgren}, Ann. Math. 84, 277--292 (1966; Zbl 0146.11905); \textit{E. Calabi}, J. Diff. Geom. 1, 111--125 (1967; Zbl 0171.20504)].\N\NTheorem. Any immersed real analytic saddle sphere \(\Sigma^2\subset \mathbb S^3\) is totally geodesic (equatorial).\N\NTo prove the statement, the authors study the distributions of principal directions on the immersed sphere and demonstrate that if \(\Sigma^2\) is not equatorial then the distributions can be well-extended through the umbilic set on \(\Sigma^2\) so that one gets two orthogonal real analytic line fields with a finite number of singularities on \(\Sigma^2\), and for any of these line fields the topological index is non-positive at each singular point. This contradicts the topological sphericity of \(\Sigma^2\).\N\NMoreover, the authors construct an example which shows that the theorem would fail if one replaced real analyticity by \(C^\infty\)-smoothness, see [\textit{G. Panina}, Adv. Geom. 5, 301--317 (2005; Zbl 1077.52003)].