Complex analysis and surface theory I
Period of duration of course
Prerequisites al consist mainly of the material covered in first-year courses. It is possible that some results proved in second-year courses may be used.
- Complex numbers
- Holomorphic functions and their main properties
- Riemann surfaces, and elements of Teichmuller theory (time allowing)
- Differential forms
- Geometry of surfaces in Euclidean space
- Isothermal coordinates
- Weierstrass representation
- Bernstein and Hopf theorems for minimal and constant mean curvature surfaces
The purpose of the course is to present elements of complex analysis, and in particular the conformal theory of holomorphic functions. This will be used in the study of the structure of surfaces in three-dimensional Euclidean space. This will help characterize some relevant classes of surfaces, such as minimal or constant mean curvature surfaces.
Useful references will be classic books on Complex Analysis, such as those by Ahlfors and Gamelin and some books concerning minimal surfaces, such as those by Osserman and Fomenko-Tuzhilin. For surface geometry classic texts are that of Do Carmo and some of Spivak's volumes.