### Descripción del curso

This course will cover in detail the recent papers of Colding and Minicozzi, *The space of embedded minimal surfaces of fixed genus in a 3-manifold I-IV*. In this work, the authors classified embedded minimal disks (EMD) in R³, and showed that the plane and the helicoid are models for any EMD, in the sense that an EMD looks like a graph (when the curvature is small) or a multi-valued graph spiralling around an axis (when the curvature is large).

The primary goal of the course is to understand the main theorems 0.1 and 0.2 in the first paper of the series. If there is enough time left, we will discuss applications of the material to the global structure of minimal surfaces, such as the uniqueness of the helicoid.

The course will be divided in 10 lessons of 2 hours each:

### Programa del Curso

**Lesson 1**: An introduction to classical theory of minimal surface. In this lecture we will show some classic results like the Theorems of Bernstein and Rado, and give the necessary background for the rest of the course. Notes can be found here.

**Lesson 2**: The Choi-Schoen theorem. Compactness of the class of minimal surfaces with fixed genus in 3-manifolds with positive Ricci curvature, cf. [9]. This is an important result that will be used later in the course. Notes can be found here.

**Lesson 3**: An overview of papers [1-4]. We will give a summary of the results proven in the four papers of Colding and Minicozzi, and we will give an idea of the arguments and the structure of the proofs. Notes can be found here.

**Lessons 4 and 5**: *The space of embedded minimal surfaces of fixed genus in a 3-manifold I: Estimates off the axis for disks*. In this paper it is shown than one can extend a small almost flat multi-valued graph up to almost the boundary of the disk. For this we use catenoid foliations to prove a curvature estimate “between the sheets”.

**Lessons 6 and 7**: *The space of embedded minimal surfaces of fixed genus in a 3-manifold II: Multi-valued graphs in disks*. In this paper it is shown that there exists a small almost flat multi-valued graph in an EMD near a point of large curvature. One proves “almost stability” of the EMD near points of large curvature, which gives curvature estimates that are used to prove the existence of small multi-valued graphs.

**Lesson 8**: *The space of embedded minimal surfaces of fixed genus in a 3-manifold III: Planar domains*. Here it is shown that near points of large curvature in an EMD there are other points of large curvature. This will give the existence of an “axis” similar to that of the helicoid, around which the multi-valued graph spirals.

**Lessons 9 and 10**: *The space of embedded minimal surfaces of fixed genus in a 3-manifold IV: Locally simply connected*. Here we prove the regularity of the axis, and the existence of a helicoid near points of large curvature of an EMD. The main tool in this paper is the “one-sided curvature estimate”, which will be shown.

Although we will go into details wherever necessary, we will assume a certain background on the theory of minimal surfaces. In particular, we assume that the audience is familiar with (or willing to accept) the first few chapters of [8].

[1] | Colding and Minicozzi, The space of embedded minimal surfaces of
fixed genus in a 3-manifold I; Estimates off the axis for
disks, Ann. Math. 160 (2004), 27-68. |

[2] | Colding and Minicozzi, The space of embedded minimal surfaces of
fixed genus in a 3-manifold II; Multi-valued graphs in disks,
Ann. Math. 160 (2004), 69-92. |

[3] | Colding and Minicozzi, The space of embedded minimal surfaces of
fixed genus in a 3-manifold III; Planar domains, Ann. Math. 160 (2004), 523-572. |

[4] | Colding and Minicozzi, The space of embedded minimal surfaces of
fixed genus in a 3-manifold IV; Locally simply connected,
Ann. Math. 160 (2004), 573-615. |

[5] | Colding and Minicozzi, Disks that are double spiral
staircases, Notices of the AMS
50 (2003), 327-339. |

[6] | Colding and Minicozzi, Embedded minimal disks, Global Theory
of minimal surfaces, 405-438, Clay Math. Proc., 2, Amer. Math. Soc. |

[7] | Colding and Minicozzi, Minimal Submanifolds, Bull. London
Math. Soc.
38 (2006), 353-395. |

[8] | Colding and Minicozzi. Minimal surfaces. Courant Lecture Notes in
Mathematics, 4. New York University, Courant Institute of
Mathematical Sciences, New York, 1999. |

[9] | Choi and Schoen. The space of minimal embeddings of a surface into a
three-dimensional manifold of positive Ricci curvature,
Invent. Math. 81 (1985), 387-394. |