Research in the Geometry of Minimal and Constant Mean Curvature Surfaces

点击翻译按钮获取中文摘要

AbstractAward: DMS 1309236, Principal Investigator: William H. Meeks The researcher will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal and constant mean curvature surfaces in three-dimensional Euclidean space, and more generally, in simply connected homogeneous three-manifolds X. One goal is to classify constant mean curvature spheres in such spaces X and to investigate when they are embedded and if they always represent solutions to isoperimetric problems in X. Another goal of the proposal is to classify the asymptotic behavior of complete embedded constant mean curvature surfaces of finite topology in three-dimensional Euclidean space, and more generally, in homogeneous three-manifolds. Theoretical techniques concerning removable singularities, compactness, regularity and convergence of sequences of embedded minimal and constant mean curvature surfaces will be investigated as well. A final goal is to understand the existence/nonexistence problem for complete, bounded embedded minimal and constant mean curvature surfaces in Euclidean three-space and in other three-dimensional spaces.Classical minimal and constant mean curvature surface theory has its roots in 18 th and 19 th century mathematics. Minimal surfaces are the first important two-dimensional examples of what is called the calculus of variations, a subject first described by Euler around 1735; constant mean curvature surfaces were first studied by Delaunay in 1835 and they also represent important examples in the calculus of variations. Physically minimal surfaces can be modeled as soap films on wires or by surfaces of least-area relative to their boundaries; physically constant mean curvature surfaces can be modeled 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 three-dimensional topology and 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.