Abstract or Additional Information
Fix a nonnegative integer g and a finite configuration of disjoint Jordan curves in Euclidean space. Then, by a classical result of Douglas, there is an area minimizer among all surfaces of genus at most g which span the given curve configuration. In the talk I will discuss a generalization of this theorem to singular configurations of possibly non-disjoint or self-intersecting curves. The proof relies on an existence result for minimal surfaces in singular metric spaces and does not seem amenable within classical smooth techniques.
This is joint work with M. Fitzi.