The closed embedded minimal surfaces in an almost positively curved three manifold
In 1985 \textit{H. I. Choi} and \textit{R. Schoen} [Invent. Math. 81, 387-394 (1985; Zbl 0577.53044)] proved that the space of compact embedded minimal surfaces of fixed topological type in a closed three-dimensional manifold with positive Ricci curvature is compact in the \(C^k\)-topology for any \(k\geq 2\). In this paper, the authors generalize this compactness theorem to the case in which the ambient manifold has positive Ricci curvature, but possesses isolated Ricci curvature zeros. After having proved a uniform lower bound for the first eigenvalue of closed embedded minimal hypersurfaces in a closed Riemannian manifold with positive Ricci curvature vanishing at isolated points, the authors, following Choi and Schoen's arguments, prove that if \(N\) is a compact three-dimensional manifold with positive Ricci curvature vanishing at isolated points and \(\{M_n\}\) is a family of closed embedded minimal surfaces of fixed genus in \(N\) uniformly outside the balls surrounding the singular points, then \(\{M_n\}\) is compact in the \(C^k\)-topology for any \(k\geq 2\).
scientific article; zbMATH DE number 792876
scientific article