The purpose of this lecture is to review some classical results on
minimal surfaces and then turn to the recent papers of Colding and
Minicozzi: * The space of embedded minimal surfaces of fixed genus
in a 3-manifold I -- IV *. We will cover as much details from
these papers as possible. The primary goal is to understand the main
theorems 0.1 and 0.2 in [4]. If there is enough time left, we
discuss applications of the material to the global structure of
minimal surfaces, such as the uniqueness of the helicoid.

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].

The lectures take place in the ** AEI, Golm ** in the ** seminar
room 0.01** on
** Mondays at 15:15**. We will try to stick to a schedule of weekly
talks.

April 7 | Jan Metzger | Classical results for minimal surfaces. Introduction to minimal surfaces, the Bernstein theorem and Rado's theorem. |

April 14 | Maria Calle | Overview of the papers [1],[2],[3],[4] |

April 21 | Tobias Lamm | The Choi-Schoen theorem. Compactness of the class of minimal surfaces with fixed genus in 3-manifolds with positive Ricci curvature, cf. [9]. |

April 28 | Jan Metzger | Colding-Minicozzi, The space of embedded minimal surfaces of fixed genus in a 3-manifold I, part 1. |

May 5 | Jan Metzger | [1], part 2. |

May 19 | Jan Metzger | [1], part 3. |

May 26 | Jan Metzger | [1], part 4. |

June 2 | Jan Metzger | [1], part 5. |

June 9 | Jan Metzger | [1], part 6. |

July 21 | Maria Calle | Colding-Minicozzi, The space of embedded minimal surfaces of fixed genus in a 3-manifold II, part 1. |

July 28 | Maria Calle | Colding-Minicozzi, The space of embedded minimal surfaces of fixed genus in a 3-manifold II, part 2. |

Possibly there will be a third talk about [2] in the week of July 28. We will then continue with papers [3] and [4] in autumn.

- Lecture 1 - Introduction and Preliminaries (PDF).
- Lecture 2 - Overview (PDF).
- Lecture 3 - The Choi-Schoen theorem (PDF).

Here is an (incomplete) list of relevant material. To get acquainted with the results and the way the arguments fit together, [5] and in particular [6] are the point to get started. Basic material concerning minimal surfaces can be found in [8] or without proofs in [7]. Links point to the versions of the papers on the arXiv.

[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. |

Last update 2008-06-11 13:16. Jan Metzger.