Course on two-dimensional Minimal Surfaces

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].

Location and Time

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.

Schedule

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.

Notes

When there are notes available in electronic format, they will be posted here.

Literature

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.