Examination procedure
Oral exam or Seminar
Syllabus
The course will provide an introduction to the theory of minimal surfaces, from their fundamental properties to some recent developments in the field. After a brief review of the necessary prerequisites, the main methods of the theory will be introduced, with particular emphasis on the interactions between minimal surfaces, global geometry, spectral theory, and topology.
Topics include:
- review of basic Riemannian geometry;
- first and second variation formulas;
- stability, the Jacobi operator, and the Morse index;
- curvature estimates and compactness theorems;
- the role of the ambient curvature in the behavior of minimal surfaces;
- relations between topology and Morse index;
- minimal surfaces in the sphere and connections with spectral optimization problems;
- an introduction to min-max methods for the existence of minimal surfaces and some applications to the geometry of 3-manifolds.
Bibliographical references
T. H. Colding and W. P. Minicozzi II, A course in minimal surfaces.
B. White, Introduction to minimal surface theory.