Geometric Analysis with Applications in Low Dimensions

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NSF Grant DMS-0204496Title: Geometric Analysis with Applications in Low DimensionsPrincipal 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 dimensionaltopology, a better understanding of certain geometric structures on3-manifolds arising from complex analysis, and a new approach to regularityissues for a type of minimizing subvariety. The first project deals withhomomorphisms from fundamental groups of Riemannian manifolds to the mappingclass group of a compact oriented surface. Examples arise as monodromyrepresentations of surface bundles. Building on previous work in this area,the PI will use harmonic map theory to give a new proof of the finitenesstheorem of Farb-Kaimanovich-Masur for lattices in higher rank Lie groups.New results for lattices in rank one groups will be obtained from thesetechniques. Symplectic Lefschetz pencils also provide especiallyinteresting examples to which this method may be applied. The second projectstudies spherical CR structures on 3-manifolds. The PI will consider threeaspects of the subject: uniformizability, rigidity, and compactness. Theresearch will develop a new approach to these problems based on subellipticanalogs of harmonic map equations. A goal of the work will be to prove acompactness theorem which will have implications for new 3-manifoldinvariants. The third part of the proposal seeks to extend the method ofTaubes to prove regularity results for certain calibrated rectifiablecurrents. This will provide a clarifying framework for a variety ofgeometric constructions that are of current interest. In addition to thesenew projects the PI will complete previous work on a conjecture of Bando-Siuconcerning the Yang-Mills flow on higher dimensional Kaehler manifolds.One of the most fascinating subjects in contemporary mathematics is thestudy of spaces of dimensions three and four. These are also the mostimportant from a physical point of view, since we live in three dimensionalspace, and dynamical behavior takes place in four dimensional space-time.The use of analytic techniques to understand the geometry and topology oflow dimensional spaces continues to be a fruitful avenue of research, butthere is much work still to be done. The focus of the PI's research in thearea of the geometry of low dimensions and mathematical applications of newideas in physics. .