Record: Yes
Level: Undergraduate
Language: English
Prerequisite
It is necessary to be familiar with the basic concepts of linear algebra and calculus. In order to be able to follow the course throughout, it is beneficial to have some basic knowledge about differential geometry or manifolds, and to be familiar with some complex analysis. However, it is also possible to make up for this within the course.
Abstract
The investigation and construction of surfaces with special geometric properties has always been an important subject in differential geometry. Of particular interest are minimal surfaces and constant mean curvature (CMC) surfaces in space forms. Global properties surfaces were first considered by Hopf, showing that all CMC spheres are round. This result was generalized by Alexandrov [2] in the 1950s, who showed that the round spheres are the only embedded compact CMC surfaces in $\mathbb R^3$, while there do not exist any compact minimal surfaces in euclidean space and hyperbolic 3-space due to the maximum principle. In contrast, there are many compact and embedded minimal surfaces in the 3-sphere, the best known examples being the Clifford torus and the Lawson surfaces, in addition to the totally geodesic 2-sphere, and a full classification is beyond the current knowledge.
One reason for the beauty and depth of minimal surface theory is that there are many different methods and tools with which to construct, study, and classify these surfaces.
In this course we will first derive basic properties of minimal surfaces in $\mathbb R^3$
and introduce some general techniques, and then move on to CMC surfaces in $\mathbb R^3$ and minimal surfaces in $\mathbb S^3.$
We mainly use differential geometric and complex analytical methods in this course.
Reference
[1] S. Brendle Minimal surfaces in S3, a survey of recent results, Bulletin of Mathematical Sciences 3, 133{171 (2013)
[2] T. Colding, W. Minicozzi, A course in minimal surfaces, Graduate Texts in Mathematics
[3] M. do Carmo, Differential Geometry of Curves and Surfaces
[4] N. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), no. 3, 627-710.
[5] H.B. Lawson, Complete minimal surfaces in S3, Ann. Math. 92(1970), 335{ 374.
Syllabus
1. minimal surfaces in $\mathbb R^3:$ Euler-Lagrange equations, curvature of surfaces in space, monotonicity formulas, examples
and partial classifications, Weierstrass formulas
2. CMC surfaces in $\mathbb R^3:$ Gauss-Codazzi equations, associated families of CMC surfaces, examples, Theorems of Alexandrov and Hopf
3. minimal surfaces in $\mathbb S^3:$ Gauss-Codazzi equations, Lawson correspondence, examples, generalized Weierstrass representations and/or classification of embedded minimal tori
Lecturer Intro
PhD in 2008, Humboldt Universitat Berlin, Germany
Habilitation in 2014, Universitat Tubingen, Germany
Researcher at Universities of Heidelberg, Hamburg, Hannover, 2014-2022
Research Fellow at Beijing Institute of Mathematical Sciences and Applications from
September 2022 on
Research interests: minimal surfaces, harmonic maps, Riemann surfaces, Higgs bundles,
moduli spaces, visualisation and experimental mathematics
Lecturer Email: seb.heller@gmail.com
TA: Dr. Houwang Li, lhwmath@bimsa.cn
