The minimal lamination closure theorem. (English) Zbl 1098.53007

Meeks, William H. III; Rosenberg, Harold

Duke Math. J. 133, No. 3, 467-497 (2006).

Summary: We prove that the closure of a complete embedded minimal surface \(M\) in a Riemannian three-manifold \(N\) has the structure of a minimal lamination when \(M\) has positive injectivity radius. When \(N\) is \(\mathbb R^3\), we prove that such a surface \(M\) is properly embedded. Since a complete embedded minimal surface of finite topology in \(\mathbb R^3\) has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi …; a complete embedded minimal surface of finite topology in \(\mathbb R^3\) is proper. More generally, we prove that if \(M\) is a complete embedded minimal surface of finite topology and \(N\) has nonpositive sectional curvature (or is the Riemannian product of a Riemannian surface with \(\mathbb R\)), then the closure of \(M\) has the structure of a minimal lamination.

Reviewer: Iskander A. Taimanov (Novosibirsk)

MSC:

53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05	Minimal surfaces and optimization
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

minimal surface; lamination

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