Mathematical Sciences: Numerical and Theoretical VariationalMethods in the Large and Applications to Geometry and Topology

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Jon Pitts will continue his study of minimal surfaces by variational techniques. This is an area which is currently experiencing dramatic developments. Pitts will focus his attention primarily on minimal surfaces in three dimensional manifolds rather than Euclidean space. The computational aspects of this research provide a new direction for Pitts' research. This is an exciting development which is very much in line with contemporary work of other researchers. The study of minimal surfaces in three dimensional manifolds is an important aspect of Thurston's geometrization conjecture. Here the investigations will concern the possibility of constructing metrics which admit infinitely many minimal spheres and tori. Another part of the project will be aimed at finding conditions on a noncompact hyperbolic manifold which guarantee that it supports a compact minimal surface. Pitts is also developing numerical methods for approximating minimal surfaces. He will implement these algorithms in software and produce graphics of the stereographic images of the surfaces. The ability to visualize these surfaces is rapidly becoming an essential ingredient of current research into these variational problems.