Current Developments in Mathematics

Volume 2008

Properly embedded minimal planar domains with infinite topology are Riemann minimal examples

Pages: 281 – 346

DOI: http://dx.doi.org/10.4310/CDM.2008.v2008.n1.a4

Authors

William H. Meeks III (University of Massachusetts, Amherst)

Joaquín Pérez (Department of Geometry and Topology, University of Granada, Spain)

Abstract

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains $M\subset \rth$ with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Pérez, and Ros states that such an $M$ must be congruent to a homothetic scaling of one of the classical examples found by Riemann in 1860. These examples ${\mathcal R}_s, 0<s<\infty $, are defined in terms of the Weierstrass ${\mathcal P}$-functions ${\mathcal P}_t$ on the rectangular elliptic curve $\frac{\C}{\langle 1, t\sqrt{-1}\rangle }$, are singly-periodic and intersect each horizontal plane in $\rth$ in a circle or a line parallel to the $x$-axis. Earlier work by Collin, López and Ros, and Meeks and Rosenberg demonstrate that the plane, the catenoid and the helicoid are the only properly embedded minimal surfaces of genus zero with finite topology (equivalently, with a finite number of ends). Since the surfaces ${\mathcal R}_s$ converge to a catenoid as $s\to 0$ and to a helicoid as $s\to \infty$, then the moduli space ${\mathcal M}$ of all properly embedded, non-planar, minimal planar domains in $\rth$ is homeomorphic to the closed unit interval $[0,1]$.

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