## Journal of Differential Geometry

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

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

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

#### Abstract

Let \mathcal M be the space of properly embedded minimal surfaces in ℝ^{3} 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 ℝ^{3} 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