In this module we study minimal hypersurfaces in Euclidean space and more generally in Riemannian manifolds. These surfaces are critical points of the area functional. We start with basic properties of minimal surfaces in flat space, like the non existence of compact minimal surfaces, the solution of the Plateau-Problem that any simple embedded curve in $\mathbb R^3$ is spanned by an area minimal surface, the theorem of Radó that a graphical boundary curve has a graphical spanning minimal area surface and the theorem of Bernstein that a minimal surface that is a graph over $\mathbb R^2$ is a flat plane.
More advanced topics include regularity properties, curvature estimates and compactnes properties, min-max constructions and the influence of the ambient geometry if the surfaces are embedded in non flat ambient manifolds.
In addition to the lecture, in the seminar part of the module we discuss modern results in this area.
- T. Colding, W. Minicozzi: A course in minimal surfaces, Graduate Studies in Mathematics, 121, AMS (2011).
- W. H. Meeks II, A. Ros: The global Theory of Minimal Surfaces in Flat Spaces, Lecture Notes in Mathematics 1775, Springer (2002).
- D. Hoffman (ed.): Global Theory of Minimal Surfaces, Clay Mathematics Proceedings 2, AMS (2001).
Basic knowlege of elliptic partial differential equations and differential geometry.
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All communications about the course are done through the Moodle platform. Here is the course link.