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:
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 |
