Mathematical Sciences: Differential Geometry and Geometric Analysis

点击翻译按钮获取中文摘要

The investigators will continue previous work on a number of projects in differential geometry and geometric analysis. Jeff Cheeger plans to gain a deeper understanding of invariants associated with certain families of elliptic operators which have recently been of some interest in Theoretical Physics. He is also proposing extensive work toward obtaining further insight into the interaction of bounded geometries, collapsing, and characteristic numbers for complete Riemanian manifolds. In addition, he will continue to expand his approximation theory for curvature of smooth manifolds by piecewise linear ones. Gromoll expects further substantial progress in his efforts to establish a global structure theory for complete Riemannian spaces of nonnegative Ricci curvature, in particular finiteness results and the construction of additional classes of examples. He also wants to proceed with his analysis of rigidity phenomena of metric foliations, i.e. locally everywhere equidistant partitions, in nonnegatively curved manifolds, which has turned out to be a crucial and delicate aspect of the global geometry of such spaces. In the area of submanifold theory he will work on various extensions of recent global results concerning isometric deformation of euclidean submanifolds, as well as explore the new theory of minimal real Kaehler submanifolds, a special but ample class of particularly interesting minimal submanifolds that surfaced in that context. Ebin plans to work in three areas all of which involve free boundary value problems. The first is the study of the motion of a thin membrane. The second is the study of dynamics of a liquid drop, and the third is the liquid drop model for a large nucleus. The second and third areas are both studied as examples of dynamical systems constrained to submanifolds of function spaces. A key step in this analysis is the study of their relative curvature. Teleman intends to continue his work on the Novikov conjecture, an outstanding problem in Differential Topology and Operator Theory. He also wants to discover a new approach to Pontrjagin classes of combinatorial manifolds, which would also be of significant interest in Theoretical Physics.