Isoliertheit und Stabilit�t von Fl�chen konstanter mittlerer Kr�mmung

In the Sobolev space Hm(B,ℝ3), B the open unit disc in ℝ2, we consider the set Mn of all conformally parametrized surfaces of constant mean curvature H with exactly n simple interior branch points (and no others). We denote by M*n the set of all xeMn with the following properties: i) in every branch point the geometrical condition KG¦xZ¦≡O holds (KG is the Gauss curvature and xz is the complex gradient of the surface x). ii) the corresponding boundary value problem Δh+×z{2(2H2-KG)h=O,hδB=O, is uniquely solvable. We prove then, that the manifold M*=UM*n is open and dense in the set of all surfaces of constant mean curvature H and that all x eM*n are isolated and stable solutions of the Plateau problem corresponding to their boundary curves. In addition, the submanifold M*n contains exactly all surfaces x for which the space of Jacobi fields is transversal (with exception of the 3-dimensional space of conformai directions) to the tangent space TxMn.