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

TL;DR

This paper discusses the classification of properly embedded minimal planar domains with infinite topology in R^3, showing they are congruent to Riemann's classical examples, and describes the structure of their moduli space.

Contribution

It proves that all such minimal domains are scaled versions of Riemann's examples, completing their classification and analyzing the moduli space structure.

Findings

Properly embedded minimal planar domains with infinite topology are congruent to Riemann's examples.

The moduli space of these domains is homeomorphic to the closed interval [0,1].

Riemann's examples interpolate between catenoid and helicoid as parameters vary.

Abstract

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, 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 {\cal R}_s, 0<s<\infty, are defined in terms of the Weierstrass {\cal P}-functions {\cal P}_t on the rectangular elliptic curve {\C} / {< 1, t\sqrt{-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, Lopez 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 {\cal R}_s converge to a catenoid as s tends to 0 and to a helicoid as s tends to infinity, then the moduli space {\cal M} of all properly embedded, non-planar, minimal planar domains in R^3 is homeomorphic to the closed unit interval [0,1].