Geometric Analysis with Applications in Low Dimensions

Johns Hopkins University, Baltimore MD

Investigators

Abstract

NSF Grant DMS-0204496 Title: Geometric Analysis with Applications in Low Dimensions Principal Investigator: Richard A. Wentworth (Johns Hopkins University) The PI proposes three research projects in the area of geometric analysis. This work will produce new results on rigidity problems in low dimensional topology, a better understanding of certain geometric structures on 3-manifolds arising from complex analysis, and a new approach to regularity issues for a type of minimizing subvariety. The first project deals with homomorphisms from fundamental groups of Riemannian manifolds to the mapping class group of a compact oriented surface. Examples arise as monodromy representations of surface bundles. Building on previous work in this area, the PI will use harmonic map theory to give a new proof of the finiteness theorem of Farb-Kaimanovich-Masur for lattices in higher rank Lie groups. New results for lattices in rank one groups will be obtained from these techniques. Symplectic Lefschetz pencils also provide especially interesting examples to which this method may be applied. The second project studies spherical CR structures on 3-manifolds. The PI will consider three aspects of the subject: uniformizability, rigidity, and compactness. The research will develop a new approach to these problems based on subelliptic analogs of harmonic map equations. A goal of the work will be to prove a compactness theorem which will have implications for new 3-manifold invariants. The third part of the proposal seeks to extend the method of Taubes to prove regularity results for certain calibrated rectifiable currents. This will provide a clarifying framework for a variety of geometric constructions that are of current interest. In addition to these new projects the PI will complete previous work on a conjecture of Bando-Siu concerning the Yang-Mills flow on higher dimensional Kaehler manifolds. One of the most fascinating subjects in contemporary mathematics is the study of spaces of dimensions three and four. These are also the most important from a physical point of view, since we live in three dimensional space, and dynamical behavior takes place in four dimensional space-time. The use of analytic techniques to understand the geometry and topology of low dimensional spaces continues to be a fruitful avenue of research, but there is much work still to be done. The focus of the PI's research in the area of the geometry of low dimensions and mathematical applications of new ideas in physics. .