Mathematical Sciences: Some Constructions of Canonical Geometric Objects

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Professor Kapouleas will study several topics in differential geometry: constant mean curvature surfaces, harmonic maps, real Einstein manifolds, and the classification of surfaces. He will use a method of "patching" to construct constant mean curvature surfaces with prescribed topological properties. The "patching" process will also be used to try to construct new examples of real Einstein manifolds. The full extent of the usefulness of the patching process is not known and one of the goals of the research is to explore other areas in which this tool might be useful. The research supported by this proposal should result in a new method of obtaining global information from local information. The standard method up to this point for obtaining global solutions to partial differential equations on surfaces involved working with the abstract space of possible global solutions to the equation. The method to be used in this research involves putting small parts of solutions together much like one puts together a jigsaw puzzle. The method has already been successful on one class of problems and the principal investigator will now try to extend the method to other classes.