Orientable and nonorientable minimal surfaces

We describe a theory of Morse-Conley type for orientable and nonorientable minimal surfaces of varying topological type solving Plateau problems in R 3 . 1991 Mathematics Subject Classification 53A10, 49F10, 58E12, 32G 15 After J. Douglas had solved Plateau's problem (this was also independently achieved by T. Rad6), he considered the problem of finding minimal surfaces of higher genus and/or connectivity bounded by a given configuration of Jordan curves in 3-space ( [D2]). In fierce competition, this problem was also studied by R. Courant and his student Shiffman ([C], [Shl]). The result stated in these papers can be expressed as follows. ~heorem 1. Let r := (rb ... , rk) be a collection of disjoint oriented Jordan curves in R 3 • Let a(r,g) := inf Area (E): E is an oriented surface of genus gin R 3 with BE =r} a*(r,g) := inf Area (E'): E' is either an oriented surface of genus< g or a union of at least two surfaces with sum of genera< g in R 3 and 8E = r}. If a(r,g) < a*(r,g), then r bounds an oriented minimal surface of genus g. The condition a(r,g) < a*(r,g) is refered to as Douglas condition. An important insight of Douglas was that instead of minimizing area it is much more convenient to minimize Dirichlet's integral 1 I 2 2) D(h, E) := 2 (hu + hv dudv