Minimal immersions of surfaces into spheres
Using the Gauss-Bonnet-Chern Theorem, the author gives a very simple proof of the following theorem, which independently was stated by \textit{V. Bangert} (unpublished) and \textit{G. R. Jensen} and \textit{M. Rigoli} [Rend. Semin. Mat., Torino, 75-99 (1983)]; Bangert's proof is the same. Theorem: Let (M,g) be a compact connected Riemannian 2-manifold and \(x: M\to S^ n(1)\) a minimal isometric immersion with Gaussian curvature K, which satisfies \(1/(\ell +1)\leq K\leq 1/\ell\) for some \(\ell \in {\mathbb{N}}\). Then there exists \(\ell \in {\mathbb{N}}\) such that either \(2\ell =k(k+1)\) or \(2\ell =k(k+1)-2\), and \(K=2/(k(k+1))\) in both cases. The value of K determines x uniquely [\textit{E. Calabi}, J. Differ. Geom. 1, 111-125 (1967; Zbl 0171.205)]. The result gives a partial answer to a conjecture of the reviewer [for further references cf. the review of \textit{T. Ogata}, Bull. Yamagata Univ., Nat. Sci. 11, 345-351 (1987; Zbl 0623.53026)].
scientific article; zbMATH DE number 4013470
scientific article