After successful completion of the course, students are able to prove the existence of convex solutions to an important class of geometric curvature problems such as Christoffel-Minkowski problems and their generalizations. 

The Minkowski problem aims at reconstructing a convex body from its surface area measure. The solution to this problem is remarkable: a Borel measure μ on the unit sphere is the surface area measure of a convex body if and only if μ has centroid at the origin and is not concentrated on a great subsphere. The course aims at providing rigorous and the most recent arguments for dealing with an important class of curvature problems which includes the Minkowski problem as a special case. The focus of this course will be on smooth solutions and obtaining a priori estimates through maximum principles (in some cases such as the Minkowski problem, a simple approximation allows us to treat Borel measures as well). The course is intended to be self-contained and presents the most recent techniques such as a viscosity approach to constant rank theorems (which deals with convexity issue), and iteration method.

Mathematical Definitions and proofs.

First lecture and organizational meeting:

Date: October 3, 3pm

 Link: https://tuwien.zoom.us/j/65272907346

Oral exam

Differentialgeometrie