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

Abstract

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M ⊂ R with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Pérez, and Ros [64] states that such an M must be congruent to a homothetic scaling of one of the classical examples found by Riemann [87] in 1860. These examples Rs, 0 < s < ∞, are defined in terms of the Weierstrass P-functions Pt on the rectangular elliptic curve C 〈1,t−1〉 , are singly-periodic and intersect each horizontal plane in R 3 in a circle or a line parallel to the x-axis. Earlier work by Collin [22], López and Ros [49] and Meeks and Rosenberg [71] 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 Rs converge to a catenoid as s → 0 and to a helicoid as s→∞, then the moduli spaceM of all properly embedded, non-planar, minimal planar domains in R is homeomorphic to the closed unit interval [0, 1]. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42.

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Cite this paper

@inproceedings{Meeks2009ProperlyEM, title={Properly embedded minimal planar domains with infinite topology are Riemann minimal examples}, author={William H . Meeks}, year={2009} }