Duke Mathematical Journal

The minimal lamination closure theorem

William H. Meeks and Harold Rosenberg

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Abstract

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 in [5, Corollary 0.7]; 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

Article information

Source
Duke Math. J. Volume 133, Number 3 (2006), 467-497.

Dates
First available in Project Euclid: 13 June 2006

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1150201199

Digital Object Identifier
doi:10.1215/S0012-7094-06-13332-X

Mathematical Reviews number (MathSciNet)
MR2228460

Zentralblatt MATH identifier
1098.53007

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Citation

Meeks, William H.; Rosenberg, Harold. The minimal lamination closure theorem. Duke Mathematical Journal 133 (2006), no. 3, 467--497. doi:10.1215/S0012-7094-06-13332-X. http://projecteuclid.org/euclid.dmj/1150201199.


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