The space of embedded minimal surfaces of fixed genus in a 3-manifold IV; Locally simply connected

Abstract

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.

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

“…By the one-sided curvature estimates of Colding and Minicozzi [8], there exists a small neighborhood U of D − {p 0 } such that the M 1,n ∩ U converge to a sublamination L p 0 ⊂ L p 0 as n → ∞, with empty singular set of convergence and such that D − {p 0 } is a limit leaf of L p 0 . After a continuation argument, the same results allow us to insure that D extends to a complete minimal surface Π in R 3 and that L p 0 extends to a possibly singular minimal lamination L of R 3 having Π as a limit leaf.…”

“…Then one can show that there exists a sequence of coaxial cylinders C n with common axis l, radii going to infinity as n → ∞ and symmetric with a certain fixed small positive height h with respect to Π, such that M 1,n ∩ C n consists only of disks for each n (because for n large, the part Ω n of M 1,n ∩ C n outside certain cone with axis l centered at p 0 consists of a highly-sheeted double multigraph over an annulus in Π, hence Ω n is topologically a disk; from here one directly obtains that M 1,n ∩ C n is a disk for n large). Using a suitable modification of the proof by Colding-Minicozzi of Theorem 0.1 in [8] with the cylinders C n replacing balls with radii going to ∞, one deduces that after passing to a subsequence, that the disks M 1,n ∩ C n converge to the foliation L Π by parallel planes of a neighborhood of Π, with singular set of convergence S(L Π ) consisting of exactly one Lipschitz curve passing through p 0 . By Meeks' regularity theorem [24], S(L Π ) is a segment contained in l. After repeating this argument at the boundary planes of L Π , we see that L Π can be enlarged to the foliation L 1 of R 3 by planes parallel to Π and that a subsequence of the M 1,n (denoted in the same way) converges to L 1 (in particular, L = L 1 ), with singular set of convergence S(L 1 ) = l. This implies M 1,n intersects B(2) in disks for n sufficiently large, which contradicts that M 1,n ∩ B(1) contains a homotopically nontrivial curve.…”

“…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.…”

On curvature estimates for constant mean curvature surfaces

“…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].…”

“…Previous important examples of curvature estimates for constant mean curvature surfaces, assuming certain geometric conditions, can be found in the literature; see for instance [2,3,4,9,10,27,28,29,30,34,35].…”