JournalsowrVol. 6 , No. 4DOI 10.4171/owr/2009/45

Minimal Surfaces

  • William H. Meeks

    University of Massachusetts, Amherst, USA
  • Matthias 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 ℝ3 or in other flat 3-manifolds. For instance, all properly embedded minimal surfaces of genus 0 in ℝ3, 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 gN, or the related Hoffman–Meeks conjecture: A finite topology surface with genus g and n ≥ 2 ends embeds minimally in ℝ3 with a complete metric if and only if ng + 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 ℝ3.
  • Minimal surfaces have had important applications in topology and play a prominent role in the larger context of geometric analysis.