The closed embedded minimal surfaces in an almost positively curved three manifold
type: Item
instance of: scholarly article
title: The closed embedded minimal surfaces in an almost positively curved three manifold
zbMATH Open document ID: 0867.53051
DOI: 10.1007/BF00773657
author: Yng-Ing Lee
author: Derchyi Wu
published in: Annals of Global Analysis and Geometry
publication date: 1995-10-09T00:00:00Z
review text: 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 \(k\geq 2\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58E12
zbMATH DE Number: 792876
zbMATH Keywords: compactness theorem
zbMATH Keywords: positive Ricci curvature
zbMATH Keywords: first eigenvalue
zbMATH Keywords: closed embedded minimal hypersurfaces
reviewed by: Ida Cattaneo Gasparini
MaRDI profile type: Publication
cites work: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature
cites work: A first eigenvalue estimate for minimal hypersurfaces
cites work: Q3344779
cites work: \(L^ p\) and mean value properties of subharmonic functions on Riemannian manifolds
cites work: Q3290297
cites work: Q3891468
cites work: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces
cites work: The existence of minimal immersions of 2-spheres
cites work: Q3336268
Wikidata QID: Q125992697
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description: scientific article; zbMATH DE number 792876