Minimal surfaces in a sphere with Gaussian curvature not less than 1/6
Let \(x: M\to S^ n(1)\) be an isometric minimal immersion of a complete, connected, oriented 2-manifold into \(S^ n(1)\), \(n\geq 3\), where the Gaussian curvature \(K\) of M satisfies \(K\geq K(r):=2/r(r+1)\). For \(r=2\) the situation was well known for a long time. For \(r=3\) the author proves that either (i) \(K\equiv 1\) \(n=2\) or (ii) \(K=1/6\) and \(n=6\) or (iii) \(n=4\). In each of the cases (i) and (ii) the immersion is uniquely determined, while in case (iii) there are infinitely many examples given by E. Tjaden [cf. \textit{M. Kozlowski} and the reviewer, Math. Z. 186, 377- 382 (1984; Zbl 0543.53045)]. For \(r=2,3\) the results partly are corollaries of results on the eigenvalues of the Laplacian [for references cf. the reviewer in \textit{L. A. Cordes} (ed.), Differential Geometry, pp. 115-120 (1985)]. The author's result is related to the reviewer's conjecture on the quantization of curvature for arbitrary r [cf. the reviewer (loc. cit.)]: The inequality K(r)\(\leq K\leq K(r-1)\) implies \(K=K(r)\) or \(K=K(r-1)\) which determines the immersion completely. \textit{V. Bangert} (private communication in Nov. 1984; see \textit{F. Dillen} [Minimal immersions of surfaces into spheres; Preprint Reihe Mathematik TU Berlin No.152 (1986)]) and \textit{G. Jensen} and \textit{M. Rigoli} [Quantization of curvature of harmonic two-spheres in complex projective space (preprint)] verify the conjecture in many cases for \(r\geq 3\). The author's result in case (iii) and p. 38-39 in Jensen's-Rigoli's preprint clarify the unprecise remark on p. 94 in [\textit{G. Jensen} and \textit{M. Rigoli}, Rend. Semin. Mat. Torino, Special Volume 75-98 (1983)].
scientific article; zbMATH DE number 3933967
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