Seminar on Minimal Surfaces Winter 2022

JProf. Peter Smillie
Fri 9-11 / Seminarraum 3
First meeting
07.10.2022 / 9.15 am / Seminarraum Statistic 02/104
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.


Useful resources

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:

Speakers and topics will be assigned at the introductory meeting.

Schedule of the lectures

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