Eine globalanalytische Behandlung des Douglas'schen Problems. (A global analysis treatment of the Douglas problem)
We consider the Douglas problem of genus 0. Starting point is the global analysis for minimal surfaces of the type of the disc which was developed by A. J. Tromba. The set of all k-connected minimal surfaces of genus 0 has a product structure, which is a consequence of the variation of the conformal type. The base space is the space of domain parameters and the fibres are the manifolds of k-connected minimal surfaces of constant conformal type. It is possible to develop a global analysis also for the more general situation considered here with the following results: The map which assigns to every minimal surface its boundary curve (in the sense of the Douglas problem) is a Fredholm operator. The index depends on the number and the total order of the branch points. The analysis allows to prove isolatedness and stability results in a relatively simple way. In the meantime the author has given a global analytic description for surfaces of genus 1, double periodic minimal surfaces, and the index theorem is proved in this case.
scientific article; zbMATH DE number 3980096