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.

Content

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.

Literature

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.

Useful resources

• 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.

Rules of the seminar

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.

Diary of the lectures

Exercises