The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces (Q1083732)

If \({\mathcal R}\) is a compact Riemann surface and \({\mathcal F}\subset {\mathcal R}^ a \)submanifold with boundary \(\partial {\mathcal F}\) such that \({\mathcal R}\setminus {\mathcal F}\) is a disk, then we prove a Riemann mapping theorem for such surfaces \({\mathcal F}\). We prove that no conformal invariant of \({\mathcal R}\) remains to be an invariant of \({\mathcal F}\). There exists a manifold of real dimension 6g-3 for any g, parametrizing the classes of conformal equivalence of such \({\mathcal F}\subset {\mathcal R}\). We describe an embedding into Teichmüller space.