# Seminar on Minimal Surfaces Autumn 2019

Lecturer
Prof. Dr. Alessandro Carlotto
Coordinator
Lectures
Mon/2w 16-18 / HG G 26.3
Tue/2w 15-17 / HG G 26.1
16.12. 16-18 / HG G 19.2
First lecture
30.09.2019
Office hours
Thu 15-17 / HG G 28 (13-15 on 14.11.)
Course Catalogue
401-3830-69L Seminar on Minimal Surfaces

This course is meant as an invitation to some key ideas and techniques in Geometric Analysis, with special emphasis on the theory of minimal surfaces. It is primarily conceived for advanced Bachelor or beginning Master students.

The minimal surface equation, examples and basic questions. Parametrized surfaces, first variation of the area functional, different characterizations of minimality. The Gauss map, basic properties. The Douglas-Rado approach, basic existence results for the Plateau problem. Monotonicity formulae and applications, including the Farey-Milnor theorem on knotted curves. The second variation formula, stability and Morse index. The Bernstein problem and its solution in the two-dimensional case. Total curvature, curvature estimates and compactness theorems. Classification results for minimal surfaces of low Morse index.

Three basic references that we will mostly refer to are the following ones:

Further, more specific references will be listed during the first two introductory lectures.

To obtain the credits for the seminar each student has to get strictly more than 70 points over 100. The points are assigned as follows:

• up to 40 points for the 40 minutes presentation;
• up to 30 points for writing the solutions to an exercise sheet, together with another student;
• up to 30 points for active participation in class.

At the end of each lecture on Tuesday, the speakers and the topics for the next two lectures (the ones held after two weeks) are assigned. For any question or doubt about the preparation of the talk, you can ask the course coordinator during the office hours (Thu 15-17 in G 28) or contact her by email to set a different time.

After each lecture, an exercise sheet concerning the topics of that lecture is published on the website. We suggest that everyone tries to solve the problems, but each student is asked to write (in LaTex, using this template) the solution to only one exercise sheet, together with another student.

Date Content Speakers Reference Notes
30.09. Introduction to the course. Minimal surface equation. Some examples and applications. Prof. A. Carlotto [Oss86] §2, [CM11] Ch.1 §1.1
01.10. Local theory of surfaces in $$\mathbb R^3$$ Prof. A. Carlotto [Oss86] §1
14.10. First variation of the area Francesco Fiorani [Oss86] §3 until pp. 23 included S.01.a
Minimal surfaces and harmonic functions Fabian Jin [Oss86] §4 until Lemma 4.2 included S.01.b
15.10. Stability inequality Carlo Schmid [CM11] Ch.1 §4 and Ch.1 §5 until Lemma 1.19 included S.02.a
Bernstein problem Daniel Paunovic [CM11] conclusion of Ch.1 §5 S.02.b
28.10. Minimal surfaces of small total curvature Martina Jorgensen [Whi16] pp. 15-17 S.03.a
The Weierstrass representation Santiago Cordero Misteli [Whi16] pp. 17-20, see also [CM11] Ch.1 §6
29.10. Monotonicity formula Philipp Reinke [Whi16] pp. 6-7 S.04.a
Density at infinity and isoperimetric inequality Michele Caselli [Whi16] pp. 7-8 and pp. 12 S.04.b
11.11. Extended monotonicity formula Ali Naseri Sadr [Whi16] pp. 8-10 until Corollary 7 included S.05.a
Farey-Milnor theorem Beat Zurbuchen [Whi16] pp. 10-12
12.11. Plateau's problem - Part I Vedran Mihal [Whi16] pp. 33-35, see also [CM11] Ch.4 §1
Plateau's problem - Part II Cédéric Hiltbrunner [Whi16] pp. 36-38, see also [CM11] Ch.4 §1
25.11. Basic compactness and curvature estimate Eleonora Eichelberg [Whi16] pp. 23-26
Concentration theorem Yann Guggisberg [Whi16] pp. 27-29 S.07.b
26.11. Stable minimal surfaces in $$\mathbb R^3$$ Quentin Biner [Whi16] pp. 30-32 S.08.a
Schoen-Yau rearrangement trick Aischa Amrhein [CM11] pp. 42-43
09.12. Choi-Schoen theorem - Part I Hjalti Ísleifsson [CM11] pp. 244 - 248 included, without proof of Prop 7.14 S.09.a
Choi-Schoen theorem - Part II Yujie Wu [CM11] pp. 249 - 251 S.09.b
10.12. Frankel's theorem Emanuel Seeman [PW03] Theorem 3 with proof
Hersch trick Giulia Docimo [CM11] Ch.7 §2
16.12. Minimal surfaces in flat tori Simone Rodoni [ACS18] Theorem 1 with proof
Assignment date Due date Exercises Solutions Solutions authors
30.09. 13.10. Exercise Sheet 1 Solution 1 Quentin Biner, Simone Rodoni
01.10. 14.10. Exercise Sheet 2 Solution 2 Eleonora Eichelberg, Yann Guggisberg
14.10. 27.10. Exercise Sheet 3 Solution 3 Aischa Amrhein, Emanuel Seeman
15.10. 28.10. Exercise Sheet 4 Solution 4 Giulia Docimo, Yujie Wu
28.10. 10.11. Exercise Sheet 5 Solution 5 Fabian Jin, Hjalti Ísleifsson
29.10. 11.11. Exercise Sheet 6 Solution 6 Francesco Fiorani, Carlo Schmid
11.11. 24.11. Exercise Sheet 7 Solution 7 Martina Jorgensen, Philipp Reinke
12.11. 25.11. Exercise Sheet 8 Solution 8 Michele Caselli, Santiago Cordero Misteli, Daniel Paunovic
25.11. 08.12. Exercise Sheet 9 Solution 9 Cédéric Hiltbrunner, Ali Naseri Sadr
26.11. 09.12. Exercise Sheet 10 Solution 10 Vedran Mihal, Beat Zurbuchen