The space of embedded minimal surfaces of fixed genus in a 3-manifolds I; Estimates off the axis for disks

Abstract

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.

The uniqueness of the helicoid

“…Let Σ be any compact smooth surface passing through the origin with boundary contained in the boundary of the ball B(1) of radius one centered at the origin. There is an ε and a constant c such that if D is an embedded minimal disk in B(1), disjoint from Σ, and with boundary contained in the boundary of B (1), then in B(ε), the curvature of D is bounded by c. This can be seen by homothetically expanding Σ; the ε depends on the norm of the second fundamental form of Σ in the ball B( 1 2 ). In our applications Σ will be a stable minimal disk for which one always has a bound on the norm of the second fundamental form in B( In this section we will prove a general structure theorem that explains some of the geometric properties that hold for a minimal lamination L of R 3 .…”

On curvature estimates for constant mean curvature surfaces

“…We show that for n sufficiently large, M n contains two, oppositely oriented 3-valued graphs on a fixed horizontal scale and with the norm of the gradient small. This is an improvement to the description in Section 2.1 where the multi-valued graphs formed on the scale of the norm of the second fundamental form; the next theorem was inspired by and generalizes Theorem II.0.21 in [6] to the nonzero constant mean curvature setting.…”

“…In the case of embedded minimal disks such a description was given by Colding and Minicozzi in [7]; see also [32,33] for related results. By rescaling arguments this description can be improved upon once one knows that the helicoid is the unique complete, embedded, non-flat minimal surface in R 3 as explained below; see [17] and also [1] for proofs of the uniqueness of the helicoid which are based in part on results in [6,7,8,9,10].…”

The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

“…Colding and Minicozzi [2] have recently applied their previous results in [5] and some new ingenious arguments to show that any complete embedded minimal surface of finite topology in R 3 is properly embedded. In particular, the conclusion of Theorem 1 remains valid if we weaken the hypothesis of properness to the hypothesis of completeness.…”

“…These papers, as well as the present one, rely on a series of deep works by Colding and Minicozzi [5,6,7,8,3] in which they attempt to describe the basic local geometry of a properly embedded minimal surface in a Riemannian three-manifold, where there is a local bound on the genus of the surface. A sequence {M (n)} n of properly embedded minimal surfaces in a Riemannian three-manifold W is called locally simply connected, if every point in W has a small neighborhood which intersects every M (n) in components which are disks with their boundary on the boundary of this neighborhood.…”