- Lecturer
- JProf. Peter Smillie

- Lectures
- Fri 9-11 / Seminarraum 3
- First meeting
- 07.10.2022 / 9.15 am / Seminarraum Statistic 02/104
- Müsli
- Seminar: Minimal Surfaces

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.

- [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. - [Web05] M. Weber
*Classical minimal surfaces in euclidean space by examples*. In Global theory of minimal surfaces, pages 19-64. American Mathematical Society, Providence, RI, edited by D. Hoffman, 2005.

- [DoC] M. do Carmo
*Differential geometry of curves and surfaces, Second Edition*. Dover Books, 2016. - Matthias Weber, Minimal surfaces blog
- J. Perez,
*A new golden age of minimal surfaces.*AMS Notices, April 2017, pp 347-358. - 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 50 points for the 40 minute presentation;
- up to 30 points for the visualization project;
- up to 20 points for active participation in class.

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 |