Arbeitsgemeinschaft: Minimal Surfaces
William H. Meeks
University of Massachusetts, Amherst, USAMatthias Weber
Indiana University, Bloomington, USA
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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 or in other flat 3-manifolds. For instance, all properly embedded minimal surfaces of genus 0 in , 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 for each , or the related Hoffman–Meeks conjecture: _A finite topology surface with genus and ends embeds minimally in with a complete metric if and only if .
- Sophisticated tools from 3-manifold theory have been applied and generalized to understand the geometric and topological properties of properly embedded minimal surfaces in .
- Minimal surfaces have had important applications in topology and play a prominent role in the larger context of geometric analysis.
Cite this article
William H. Meeks, Matthias Weber, Arbeitsgemeinschaft: Minimal Surfaces. Oberwolfach Rep. 6 (2009), no. 4, pp. 2545–2584
DOI 10.4171/OWR/2009/45