Commentarii Mathematici Helvetici

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Volume 80, Issue 4, 2005, pp. 811–858
DOI: 10.4171/CMH/36

The theory of minimal surfaces in $M \times \mathbb{R}$

William H. Meeks[1] and Harold Rosenberg[2]

(1) Mathematics Department, University of Massachusetts, MA 01003, AMHERST, UNITED STATES
(2) IMPA, Estrada Dona Castorina 110, 22460-320, RIO DE JANEIRO, BRAZIL

In this paper, we develop the theory of properly embedded minimal surfaces in $M \times \mathbb{R}$, where $M$ is a closed orientable Riemannian surface. We construct many examples of different topology and geometry. We establish several global results. The first of these theorems states that examples of bounded curvature have linear area growth, and so, are quasiperiodic. We then apply this theorem to study and classify the stable examples. We prove the topological result that every example has a finite number of ends. We apply the recent theory of Colding and Minicozzi to prove that examples of finite topology have bounded curvature. Also we prove the topological unicity of the embedding of some of these surfaces.

Keywords: Minimal surface, flux, stable minimal surface, index of stability, minimal lamination, curvature estimates, periodic minimal surface, quasiperiodic

Meeks William, Rosenberg Harold: The theory of minimal surfaces in $M \times \mathbb{R}$. Comment. Math. Helv. 80 (2005), 811-858. doi: 10.4171/CMH/36