Research in the Geometry of Minimal and Constant Mean Curvature Surfaces

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Abstract of Grant ProposalIn this proposal the researcher will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal and constant mean curvature surfaces in the 3-dimensional Euclidean space and in a general simply-connected homogeneous 3-manifolds X. There are several goals of the proposal which include: (1) To classify the asymptotic behavior of complete embedded constant mean curvature surfaces of finite topology in 3-dimensional Euclidean space, and more generally, in a general X. (2) To prove that the moduli space M(X) of constant mean curvature spheres in X is an interval parametrized by the mean curvature function and consists entirely of index-1 spheres (this interval has an end point with an index-0 sphere in the case X the the product of a round sphere with R). (3) When X is homeomorphic to 3-dimensional Euclidean space, then the spheres in M(X) are embedded and can be chosen to give rise to a foliation of the space X punctured in a single point. (4) When X is homeomorphic to 3-dimensional Euclidean space, then solutions to the isoperimetric problems in X are unique and bounded by the spheres in M(X). Related theoretical techniques concerning compactness, removable singularities results, regularity and convergence of embedded minimal and constant mean curvature surfaces of locally bounded genus will be investigated as well. Another main goal is to understand the existence and nonexistence of complete, bounded embedded minimal and constant mean curvature surfaces in Euclidean 3-space and in other 3-dimensional manifolds.Classical minimal and constant mean curvature surface theory has its roots in18-th and 19-{th century mathematics. Minimal surfaces are the first important two dimensional examples of what is called the calculus of variations, first described by Euler around 1735; constant mean curvature surfaces were first studied by Delaunay in 1835 also represent important examples in the calculus of variations. Physically minimal surfaces can be modeled locally as soap films on wires or by surfaces of least-area relative to their local boundaries; physically constant mean curvature surfaces can be modeled locally as soap films on wires with a constant pressure difference on each side. Minimal surfaces play an important role as a tool in the study of 3-dimensional topology and Riemannian geometry. The research in this proposal concerns global and local properties of embedded minimal and constant mean curvature surfaces and possible applications of these results to basic research in three-dimensional topology and geometry. In part because of the important connections with other areas of mathematics and the ease in which it is possible to make computer graphics pictures of beautiful classical examples, minimal and constant mean curvature surfaces continue to be one of the principal topics for popular science articles and public science exhibits. Thus, indirectly, the exciting research problems outlined in this proposal help bring many young scientists and mathematicians to the frontiers of research.