This course is an introduction to the theory of minimal surfaces, focusing on minimal surfaces in R^3. We will cover both classical complex analytic methods and modern geometric analytic methods. We will supplement the theory with many examples, which we will 3D print.
The theory portion of the course will cover: The minimal surface equation, examples and basic questions. The first variation formula and the monotonicity formula, with application to the Farey-Milnor theorem on knotted curves. The Gauss map, Osserman's finite total curvature theorem. Curvature estimates and compactness theorems. The second variation formula, stability, and Morse index. The Plateau problem and the Douglas-Rado theorem. Classification problem for complete minimal surfaces in R^3. The Weierstrass representation, and the explicit formulas of Schwarz and Bjorling.
In the visualization part of the course, we will use the Weierstrass representation to study: The classical minimal surfaces, periodic minimal surfaces, Bjorling surfaces, complete embedded minimal surfaces of low genus, adding genus and degeneration of families.
The intended prerequisite is basic differential geometry of curves and surfaces in space, for instance as covered by Do Carmo, Differential Geometry of Curves and Surfaces, chapters 1-4. It will also be helpful to know some basic complex analysis, for instance as covered by Ahlfors, Complex Analysis, chapters 1-4.
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:
Speakers and topics will be assigned at the introductory meeting.
| Date | Content | Speakers | Reference | Notes | |
|---|---|---|---|---|---|
| 07.10. | Introductory meeting. Pictures, questions, and organization. | Peter Smillie | |||
| 21.10. | Local theory of surfaces in R^3 | Peter Smillie | [Oss86] §1,§2, [CM11] Ch.1 §1.1, [DoC] Ch.2 | Exercises 1 | |
| 28.10. | Properties of curvature | Peter Smillie | [Oss86] §1,§2, [CM11] Ch.1 §1.1, [DoC] Ch.3 | Exercises 1 | |
| 04.11. | First variation of the area | Peter Smillie | [Oss86] §3, [Whi16] §1.0 and §1.4 | Exercises 3 | |
| 11.11. | Harmonicity | Peter Smillie | [Oss86] §4, [Whi16] §2.1 and §2.4, [CM11] Ch.1 §6 | ||
| 18.11. | Review | Peter Smillie | |||
| 25.11. | Weierstrass representation | Peter Smillie | |||
| 02.12. | Monotonicity and Fary-Milnor | Marvin Hertweck | [Whi16] §1.1 - §1.3 | ||
| 9.11 | Finite total curvature theorem | Rico Görlach | [Whi16] §2.3 | ||
| 16.12 | Curvature estimates and compactness | Theresa Mohr | [Whi16] §3.0-§3.3 | ||
| 13.01. | Plateau's problem | Florent Schaffhauser | [Whi16] pp. 33-38, see also [CM11] Ch.4 §1 | ||
| TBD | Branch points and embeddednes of Plateau | TBD | [Whi16] pp. 38-39, [CM11] Ch 6 §1,2 | ||
| TBD | Stability and Bernstein's theorem | TBD | [Whi16] §3.5, [CM11] Ch.1 §4 and Ch.1 §5 | ||
| 10.02. | Visualization project final presentations |