The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks
By T. Colding, W. Minicozzi. 2002
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This paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 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 . We show here that if the curvature of such a disk becomes large at some point, then it contains an almost flat multi-valued graph nearby that continues almost all the way to the boundary. This will be proved by showing the existence of small multi-valued graphs near points of large curvature and then using the extension result for multi-valued graphs proved in the first paper in this series. 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 like a helicoid. Recall that a double spiral staircase consists of two spiral staircases that spiral together around a common axis, one inside the other. 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 it is a double spiral staircase, we will first prove that it is built out of N-valued graphs where N is a fixed number. These N-valued graphs are like a single spiral staircase connecting N floors. The existence of the N-valued graphs was initiated in the first paper and will be completed here. 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.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/3ad9abd24a49c3ae1ee2cd7700821ce2ccd4f20e
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- The Geometry of Minimal Surfaces of Finite Genus I: Curvature Estimates and Quasiperiodicity (2004, background): “can be analyzed by means of the local and global compactness and regularity theorems by Colding and Minicozzi [2, 3 , 5]. Our proofs depend heavily on their technical results.”
- 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.”
- The uniqueness of the helicoid (2005, background): “2 of [2] for the multigraphs M(ij) that begin around S(L) near the origin.”
- The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus (2005, methodology): “This is because the arguments of [CM3]–[CM6] for disks generalize to surfaces with bounded genus and one end; in fact, many of these results were stated in that generality (see, e.g., theorem 1.22 in [ CM4 ]).”
- The classical theory of minimal surfaces (2011, background): “Colding and Minicozzi obtained Corollary 9.5 from a deep analysis of the relation between intrinsic and extrinsic distances for an embedded minimal disk in R3 (see [37]), and the proof of Theorem 9.2 is inspired by this previous work of Colding and Minicozzi.”
- Conformal properties in classical minimal surface theory (2004, unknown): “The recent ground breaking work by Colding and Minicozzi in [24, 15, 16 , 17, 23, 14] has been especially influential.”