Isoliertheit und Stabilität von Flächen konstanter mittlerer Krümmung

Abstract

In the Sobolev space H^m(B,ℝ³), B the open unit disc in ℝ², we consider the set M_n 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 xεM_n with the following properties:

• i) in every branch point the geometrical condition K_G¦x_Z¦≡O holds (K_G is the Gauss curvature and x_z is the complex gradient of the surface x).

• ii) the corresponding boundary value problem Δh+×_z{²(2H²-K_G)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 εM*_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 T_xM_n.