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

By T. Colding, W. Minicozzi. 2002

162 citations recorded by the source corpus.

This 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 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3. 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3fcbdfab6155ad8b5ed1fba5651fd2abda3ed838

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Conformal properties in classical minimal surface theory (2004, background): “The recent ground breaking work by Colding and Minicozzi in [19, 14 , 13, 15, 18] has been especially inuen tial.”
The classical theory of minimal surfaces (2011, background): “1 in Colding and Minicozzi [34] for the proof of this result when the leaves are twodimensional).”
The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus (2005, methodology): “Part I is devoted to recalling some of the earlier results for disks given in [CM04b]–[CM04e] and [CM08].”
The minimal lamination closure theorem (2006, methodology): “Theorem 2 implies Colding and Minicozzi’s recent theorem that a complete embedded minimal surface of finite topology in R3 is properly embedded.”
Stable constant mean curvature surfaces (2008, methodology): “In the minimal case, this result was found independently by Colding and Minicozzi [7] and it is motivated by an earlier almost identical result in [38] and by still earlier work of Gulliver and Lawson [24].”
The theory of minimal surfaces in M × R (2005, unknown): “In our recent paper [24], we generalize, to the three-manifold setting, some of the key theorems of Colding and Minicozzi in [4], where they prove that a complete embedded minimal surface of finite topology in R3 is properly embedded.”
The Geometry of Minimal Surfaces of Finite Genus I: Curvature Estimates and Quasiperiodicity (2004, methodology): “In a recent series of papers, Colding and Minicozzi [2, 3, 4, 5 , 6] have attempted to describe the basic structure of compact embedded minimal surfaces M of fixed genus which are contained in the unit ball B = B(~0,1) and which have their boundary on the boundary of B. The most important case of t”