Arbeitsgemeinschaft: Minimal Surfaces

Abstract

The theory of Minimal Surfaces has developed rapidly in the past 10 years. There are many factors that have contributed to this development:

• Sophisticated construction methods [14,29,31] have been developed and have supplied us with a wealth of examples which have provided intuition and spawned conjectures.
• Deep curvature estimates by Colding and Minicozzi [3] give control on the local and global behavior of minimal surfaces in an unprecedented way.
• Much progress has been made in classifying minimal surfaces of finite topology or low genus in $R^3$ or in other flat 3-manifolds. For instance, all properly embedded minimal surfaces of genus 0 in $R^3$, even those with an infinite number of ends, are now known [21, 23, 25].
• There are still numerous difficult but easy to state open conjectures, like the genus-g helicoid conjecture: _There exists a unique complete embedded minimal surface with one end and genus $g$ for each $g\in \mathbb{N}$, or the related Hoffman–Meeks conjecture: _A finite topology surface with genus $g$ and $n\ge 2$ ends embeds minimally in $R^3$ with a complete metric if and only if $n\le g+2$.
• Sophisticated tools from 3-manifold theory have been applied and generalized to understand the geometric and topological properties of properly embedded minimal surfaces in $R^3$.
• Minimal surfaces have had important applications in topology and play a prominent role in the larger context of geometric analysis.