## A New Golden Age of Minimal Surfaces

- Michael J. Barany, Svitlana Mayboroda, +4 authors Vaughn Climenhaga
- 2017

2 Excerpts

- Published 2009

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.

Showing 1-10 of 83 references

Highly Influential

20 Excerpts

Highly Influential

20 Excerpts

Highly Influential

20 Excerpts

Highly Influential

4 Excerpts

Highly Influential

8 Excerpts

Highly Influential

5 Excerpts

Highly Influential

4 Excerpts

Highly Influential

13 Excerpts

Highly Influential

15 Excerpts

Highly Influential

20 Excerpts

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