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.
Cite this article
William H. Meeks, Matthias Weber, Minimal Surfaces. Oberwolfach Rep. 6 (2009), no. 4, pp. 2545–2584DOI 10.4171/OWR/2009/45