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Volume 6, Issue 4, 2009, pp. 2545–2584
Published online: 2010-09-01
William H. Meeks
and Matthias Weber
(1) University of Massachusetts, Amherst, USA
(2) Indiana University, Bloomington, USA
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  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 g ∈ N, 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 n ≤ 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 ℝ3.
- Minimal surfaces have had important applications in topology and play a prominent role in the larger context of geometric analysis.
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