Journal of Differential Geometry

The Geometry of Minimal Surfaces of Finite Genus I: Curvature Estimates and Quasiperiodicity

William H. Meeks III, Joaquín Pérez, and Antonio Ros

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Abstract

Let \mathcal M be the space of properly embedded minimal surfaces in ℝ3 with genus zero and two limit ends, normalized so that every surface M ∊ \mathcal M has horizontal limit tangent plane at infinity and the vertical component of its flux equals one. We prove that if a sequence {M(i)} i ∊ \mathcal M has the horizontal part of the flux bounded from above, then the Gaussian curvature of the sequence is uniformly bounded. This curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences, concerning the geometry of any properly embedded minimal surface in ℝ3 with finite genus, and the possible limits through a blowing-up process on the scale of curvature of a sequence of properly embedded minimal surfaces with locally bounded genus in a homogeneously regular Riemannian 3-manifold.

Article information

Source
J. Differential Geom., Volume 66, Number 1 (2004), 1-45.

Dates
First available in Project Euclid: 21 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090415028

Digital Object Identifier
doi:10.4310/jdg/1090415028

Mathematical Reviews number (MathSciNet)
MR2128712

Zentralblatt MATH identifier
1068.53012

Citation

Meeks III, William H.; Pérez, Joaquín; Ros, Antonio. The Geometry of Minimal Surfaces of Finite Genus I: Curvature Estimates and Quasiperiodicity. J. Differential Geom. 66 (2004), no. 1, 1--45. doi:10.4310/jdg/1090415028. https://projecteuclid.org/euclid.jdg/1090415028


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