The space of embedded minimal surfaces of fixed genus in a 3-manifold. V. Fixed genus
The following compactness questions naturally arise when considering a class of submanifolds of a fixed complete Riemannian manifold \((M, g)\): (1) Given a sequence \(\Sigma_i\) of smoothly immersed submanifolds, to what extent can we say that, up to passing to a subsequence, the \(\Sigma_i\) converge to a limit \(\Sigma\)? (2) Moreover, assuming that all the \(\Sigma_i\) have a property \(\mathcal{P}\), to what extent does this affect the answer to the previous question and to what extent does \(\Sigma\) have the property \(\mathcal{P}\)? Answering this type of questions has important consequences because, often, when studying a class of submanifolds useful information can be obtained about the class by considering to what extent the class is compact and by analyzing how failure of compactness may happen. The paper under review is the fifth of a series of papers containing Colding and Minicozzi's recent work regarding compactness properties of embedded minimal surfaces in three-manifolds. They mainly study the failure of smooth convergence for sequences of embedded minimal surfaces in \(\mathbb{R}^3\) with bounded genus because, as they see, the key is to understand the structure of an embedded minimal planar domain in a ball in \(\mathbb{R}^3\), that is, the case when the surfaces have genus zero, since the general case of fixed genus requires only minor changes. In the first four papers, they considered the case of minimal disks. Roughly speaking, they characterized the failure of smooth compactness for embedded minimal disks in \(\mathbb{R}^3\), with the failure being always modelled on the helicoid. Their proof utilized and combined many properties of embedded minimal surfaces in \(\mathbb{R}^3\) and, in particular, they proved the so-called one-sided curvature estimate, which gives a uniform curvature bound for an embedded minimal disk that is close to, and on one side of, a plane. The focus in this paper is on non-simply connected planar domains. Sequences of planar domains which are not simply connected are, after passing to a subsequence, naturally divided into two separate cases depending on whether or not the topology is concentrating at points. These cases are distinguished as follows: A sequence of surfaces \(\Sigma_i^2\subset \mathbb{R}^3\) is called {\textit{uniformly locally simply connected}} (or ULSC) if for each compact subset \(K\) of \(\mathbb{R}^3\), there exists a constant \(r_0 > 0\) (depending on \(K\)) so that for every \(x \in K\), all \(r \leq r_0\) and every surface \(\Sigma_i\), each connected component of \(B_{r}(x) \cap \Sigma_i\) is a disk. A point \(y\) in \(\mathbb{R}^3\) is said to be in \(\mathcal{S}_{\mathrm{ulsc}}\) if the curvature for the sequence \(\Sigma_i\) blows up at \(y\) and the sequence is ULSC in a neighborhood of \(y\). The authors first show that every sequence \(\Sigma_i\) has a subsequence that is either ULSC or for which \(\mathcal{S}_{\mathrm{ulsc}}\) is empty and then prove that these two different cases give two very different structures. Following their earlier papers on disks, they obtain two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The first of these says that any such surface without small necks can be obtained by gluing together two oppositely-oriented double spiral staircases, while the second theorem gives a pair of pants (a topological disk with two subdisks removed) decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves. Both of these structures occur as different extremes in the \(2\)-parameter family of minimal surfaces known as the Riemann examples.
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