- Lecturer
- Prof. Dr. Alessandro Carlotto
- Coordinator
- Giada Franz

- Lectures
- Thu 16-18 / HG D 3.2
- First lecture
- 24.02.2022
- Course Catalogue
- 401-3830-22L Seminar on Minimal Surfaces (an Invitation to Geometric Analysis)

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:

- [Whi16] B. White,
*Introduction to minimal surface theory*. Geometric analysis, 387-438, IAS/Park City Math. Ser., 22. American Mathematical Society, Providence, RI, 2016. - [CM11] T. Colding, W. Minicozzi,
*A course in minimal surfaces*. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. - [Oss86] R. Osserman,
*A survey of minimal surfaces*. Second edition. Dover Publications, Inc., New York, 1986. vi+207 pp. - [PW03] P. Petersen, F. Wilhelm,
*On Frankel's theorem*. Canad. Math. Bull. 46 (2003), no. 1, 130–139. - [ACS18] L. Ambrozio, A. Carlotto, B. Sharp,
*A note on the index of closed minimal hypersurfaces of flat tori*. Proc. Amer. Math. Soc. 146 (2018), no. 1, 335–344.

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

- Notes about the weak maximum principle and the strong maximum principle (from Functional Analysis II - Spring 2021).
- T. Tao, Talks are not the same as papers.
- J. Ellenberg, Tips on giving talks.
- MIT OCW, Preparing 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 Thursday, the speakers and the topics for the next lecture are assigned. For any question or doubt about the preparation of the talk, you can ask the course coordinator during the office hours (Mon 15-17 in HG 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 |
---|---|---|---|---|

24.02. | Introduction to the course. Minimal surface equation. Some examples and applications. | Prof. A. Carlotto | [Oss86] §2, [CM11] Ch.1 §1.1 | |

03.03. | Local theory of surfaces in \(\mathbb R^3\) | Prof. A. Carlotto | [Oss86] §1 | |

10.03. | First variation of the area | Niklas Canova | [Oss86] §3 until pp. 23 included | |

Minimal surfaces and harmonic functions | Giada Franz | [Oss86] §4 until Lemma 4.2 included | ||

17.03. | Stability inequality | Aurel Zürcher | [CM11] Ch.1 §4 and Ch.1 §5 until Lemma 1.19 included | |

Bernstein problem | Jason Brüderlin | [CM11] conclusion of Ch.1 §5 | ||

24.03. | Minimal surfaces of small total curvature | Greg Weiler | [Whi16] pp. 15-17 | |

The Weierstrass representation | Huaitao Gui | [Whi16] pp. 17-20, see also [CM11] Ch.1 §6 | Examples | |

31.03. | Monotonicity formula | Cosmin Manea | [Whi16] pp. 6-8 | |

Extended monotonicity and Farey-Milnor theorem | Gabriel Frey | [Whi16] pp. 8-12 | ||

07.04. | Plateau's problem | Marek Kurczynski | [Whi16] pp. 33-38, see also [CM11] Ch.4 §1 | |

28.04. | Basic compactness and curvature estimate | Julian Bürge | [Whi16] pp. 23-26 | |

Concentration theorem | Tristan Strumann | [Whi16] pp. 27-29 | ||

05.05. | Second variation of the area | Giada Franz | [CM11] Ch.1 §8 | |

12.05. | Choi-Schoen theorem - Part I | Chiara Tschopp | [CM11] pp. 244 - 248 included, without proof of Prop 7.14 | |

Choi-Schoen theorem - Part II | Filippo Paiano | [CM11] pp. 249 - 251 | ||

19.05. | Frankel's theorem | Elia Mazzucchelli | [PW03] Theorem 3 with proof | |

Hersch trick | Tim Haupt | [CM11] Ch.7 §2 |

Assignment date | Due date | Exercises | Solutions | Solutions authors | Extras |
---|---|---|---|---|---|

24.02. | 09.03. | Exercise sheet 1 | Solution 1 | Tristan Strumann, Chiara Tschopp | |

03.03. | 16.03. | Exercise sheet 2 | Solution 2 | Julian Bürge, Tim Haupt | Mathematica code |

10.03. | 23.03. | Exercise sheet 3 | Solution 3 | Marek Kurczynski, Filippo Paiano | |

17.03. | 30.03. | Exercise sheet 4 | Solution 4 | Elia Mazzucchelli | |

24.03. | 06.04. | Exercise sheet 5 | Solution 5 | Niklas Canova, Aurel Zürcher | |

07.04. | 27.04. | Exercise sheet 6 | Solution 6 | Cosmin Manea | |

28.04. | 11.05. | Exercise sheet 7 | Solution 7 | Jason Brüderlin, Greg Weiler | |

05.05. | 18.05. | Exercise sheet 8 | Solution 8 | Gabriel Frey, Huaitao Gui |