Mathematical Sciences: Complete Surfaces in Three-Space

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Brian Smyth and Frederico Xavier will continue their work on the theory of complete minimal surfaces. This is a subject where great advances have taken place in the last decade. These were to a large extent based on the conformal character of such surfaces. In the current project attention will be focussed on more general surfaces. Many of the important unresolved problems in this area depend on an understanding of the analytic properties of the Gauss map. In particular it is important to investigate which meromorphic functions on the unit disc arise as Gauss maps of complete minimal surfaces. One of the principal motivations for this work arises from Efimov's result that any complete surface in three dimensional space which has negative Gauss curvature must have points where this curvature is arbitrarily close to zero. This in turn is related to the conjecture of Caratheodory that every compact surface in three dimensional space which has positive Gauss curvature must have at least two umbilic points. Smyth and Xavier have already made substantial progress towards this conjecture and will continue their efforts in this direction. Other problems which will be investigated include the question as to whether there are bounded non-compact complete surfaces with negative Gauss curvature.