Properly embedded minimal planar domains

Abstract

In 1997, Collin proved that any properly embedded minimal surface in $\mathbb{R}^3$ with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are the only non-planar, non-simply connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ of finite topology. In 2005, Meeks and Rosenberg proved that the only simply connected, properly embedded minimal surfaces in $\mathbb{R}^3$ are planes and helicoids. Around 1860, Riemann defined a one-parameter family of periodic, infinite topology, properly embedded, minimal planar domains $\mathcal{R}_t$ in $\mathbb{R}^3$, $t\in (0,\infty )$. These surfaces are called the Riemann minimal examples, and the family $\{ \mathcal{R}_t\} _t$ has natural limits being a vertical catenoid as $t\to 0$, and a vertical helicoid as $t\to \infty $. In this paper we complete the classification of properly embedded, minimal planar domains in $\mathbb{R}^3$ by proving that the only connected examples with infinite topology are the Riemann minimal examples. We also prove that the limit ends of Riemann minimal examples are model surfaces for the limit ends of properly embedded minimal surfaces $M\subset \mathbb{R}^3$ of finite genus and infinite topology, in the sense that such an $M$ has two limit ends, each of which has a representative which is naturally asymptotic to a limit end representative of a Riemann minimal example with the same associated flux vector.