Arbeitsgemeinschaft: Minimal Surfaces

  • William H. Meeks

    University of Massachusetts, Amherst, USA
  • Matthias Weber

    Indiana University, Bloomington, USA
Arbeitsgemeinschaft: Minimal Surfaces cover
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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