Absolute uniqueness of minimal surfaces bounded by contours with a one-to-one projection onto a plane

In this paper the uniqueness of the always existing Douglas-Radó solution to Plateau’s Problem of finding a minimal surface of the type of the disc to a given rectifiable Jordan Curve Γ in ?³ is considered in combination with the question: When is the solution surface also a graph? So only contours with a one-to-one projection onto a plane are candidates. The uniqueness is generalized to all genera and orientations and the non orientable case and then called “absolute”. It is demonstrated, that if it we have a continuous family of catenoids as support surfaces or barriers at the non convex projected part of the C ² contour, assured by the non negativity of a function of the radii of the necks of the catenoids and the contour, the Douglas-Radó solution is absolute unique and a graph. Especially there are no further restrictions to the projection of the contour than C ²-smoothness. As easy first consequences, the earlier result of Sauvigny, using Scherk’s surface as support surfaces for certain contours with a non convex projection, is generalized to absolute uniqueness and the problem initiating, famous uniqueness and existence result in case of a convex projection of Radó, Kneser and Meeks is also carried over to absolute uniqueness. Finally, by a generic example using our mentioned main fruit of investigation, the non necessity of any curvature bound, any bound on function norms, any height bound, and Williams local tangential Lipschitz condition for the whole contour for the existence of a solution for the minimal surface equation or uniqueness of parametric minimal surfaces is shown. This covers more than all known sufficient conditions, i.e. the well known 4π curvature bound of Nitsche, Sauvigny’s uniform concavity-, Lau’s smallness in the C ^0,1-norm condition and Meeks restriction of the deviation from the plane in the C ²-norm. The article is strongly geometrical in spirit so enabling the use of geometrical imagination and intuition in the realms of results and techniques from many different, including very advanced and abstract, mathematical fields. As the proofs are very complete, transparent and proceed in little steps, covering also new elementary results, the article is predestined for educational employment. Especially for its use of the maximum principle in a very sophistcated, long run, merged with topology, way and the new setting of surfaces with a Riemannian structure and usual derivations for handling the non-orientable case.