It has long been conjectured that starting at a generic smooth closed embedded surface in R 3 , the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.

IntroductionThis paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key is to understand the structure of an embedded minimal disk in a ball in R 3 . This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details.Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the one-sided curvature estimate). The lamination theorem is stated in the global case where the lamination is, in fact, a foliation. The first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph. This is done by showing that if the curvature is large at some point (and hence the surface is not a graph), then it is a double spiral staircase like the helicoid. To prove that such a disk is a double spiral staircase, we showed in the first three papers of the series that it is built out of N -valued graphs where N is a fixed number. In this paper we will deal with how the multi-valued graphs fit together and, in particular, prove regularity of the set of points of large curvature -the axis of the double spiral staircase.The first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase.

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IntroductionThis paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R 3 (with the flat metric). This study is undertaken here and completed in [CM6]. These local results are then applied in [CM7] where we describe the general structure of fixed genus surfaces in 3-manifolds.There are two local models for embedded minimal disks (by an embedded disk, we mean a smooth injective map from the closed unit ball in R 2 into R 3 ). One model is the plane (or, more generally, a minimal graph), the other is a piece of a helicoid. In the first four papers of this series, we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multi-valued graph. This will be done by showing that if the curvature is large at some point (and hence the surface is not a graph), then it is a double spiral staircase. To prove that such a disk is a double spiral staircase, we will first prove that it is built out of N -valued graphs where N is a fixed number. This is initiated here and will be completed in the second paper. The third and fourth papers of this series will deal with how the multi-valued graphs fit together and, in particular, prove regularity of the set of points of large curvature -the axis of the double spiral staircase.The reader may find it useful to also look at the survey [CM8] and the expository article [CM9] for an outline of our results, and their proofs, and how these results fit together. The article [CM9] is the best to start with.

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces (i.e., surfaces without self-intersection). In fact, we will prove considerably more. The heart of our argument is very general and should apply to a variety of situations, as will be more apparent once we describe the main steps of the proof later in the introduction.The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed versions were false; we will show here that for embedded surfaces, i.e., injective immersions, they are in fact true.Their original form was given in 1965 in [Ca] where E. Calabi made the following two conjectures about minimal surfaces (they were also promoted by S.S. Chern at the same time; see page 212 of [Ch]): Conjecture 0.1. "Prove that a complete minimal hypersurface in R n must be unbounded." Calabi continued: "It is known that there are no compact minimal submanifolds of R n (or of any simply connected complete Riemannian manifold with sectional curvature ≤ 0). A more ambitious conjecture is": Conjecture 0.2. "A complete [non-flat] minimal hypersurface in R n has an unbounded projection in every (n − 2)-dimensional flat subspace."