On the Horgan minimal non-surface (Q1272701)

The idea of the Horgan surface was that of a minimal surface obtained by adding another handle to the minimal surface of C.~Costa, an embedded surface of the topological type of a plane with a punctured handle. It occured as a heuristical result of computer experiments and visualizations of D.~Hoffman, H.~Karcher and the author in the early 1990's, and got his name from a discussion raised by an article entitled ``Death of Proof'' by J. Horgan that appeared 1993 in 'Scientific American', claiming that in mathematics in contrast to rigorous proofs, computer aided proofs, experimental methods and visual methods, as they are used for instance in chaos theory and nonlinear dynamics, are of growing importance. The question remained open, whether there exists an embedded minimal surface of this topological type, i. e., with one planar and two catenoidal ends, of genus two, and of finite total curvature. They produced pictures of such a surface that could make the reader believe, to see a minimal surface. The aim of this paper is to explain how such pictures are generated and to prove, that no minimal surface with these properties can exist.