The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. (English) Zbl 0577.53044
Let N be a compact 3-dimensional Riemannian manifold with positive Ricci curvature. It is proved that there is a constant C depending only on N and an integer \(\chi\) such that if M is a compact embedded minimal surface in N with Euler characteristic \(\chi\), then max \(\| A\| \leq C\), where \(\| A\|\) is the length of the second fundamental form. The proof is based on the following compactness theorem: The space of compact embedded minimal surfaces of fixed topological type in N is compact in the \(C^ k\) topology for any \(k\geq 2\). Furthermore, if N is real analytic, then this space is a compact finite-dimensional real analytic variety.
Reviewer: G.Thorbergsson
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
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