Geometric Analysis with Applications in Low Dimensions

NSF Grant DMS-0204496题目：几何分析及其在低维中的应用首席研究员：理查德·A·温特沃斯(约翰·霍普金斯大学)PI在几何分析领域提出了三个研究项目。这项工作将在低维拓扑中的刚性问题上产生新的结果，更好地理解由复分析产生的三维流形上的某些几何结构，以及一种最小化子簇的正则性问题的新方法。第一个项目研究从黎曼流形的基本群到紧致定向曲面的映射类群的同态。例如，曲面丛的单行表示。本文在前人工作的基础上，利用调和映射理论，对高阶李群中的格的Farb-Kaimanovich-Masur的有限性定理给出了一个新的证明，并由此得到了一阶群中格的新结果。辛Lefschetz铅笔也提供了这种方法可以应用的特别有趣的例子。第二个项目研究了三维流形上的球面CR结构。PI将考虑主题的三个方面：一致性、刚性和紧凑性。这一研究将发展一种基于调和映射方程次椭圆化模拟的新方法。这项工作的一个目标是证明一个紧性定理，它将对新的3-流形不变量产生影响。提案的第三部分试图扩展Taubes的方法，以证明某些已校准的可校正电流的正则性结果。这将为当前感兴趣的各种几何结构提供一个明确的框架。除了这些新的项目外，PI还将完成Bando-Siu关于高维Kaehler流形上的Yang-Mills流的猜想的先前工作。当代数学中最吸引人的学科之一是研究三维和四维空间。从物理的角度来看，这些也是最重要的，因为我们生活在三维空间中，而动力学行为发生在四维时空中。使用解析技术来理解低维空间的几何和拓扑仍然是一条卓有成效的研究途径，但仍有许多工作要做。PI在低维几何和Newideas在物理中的数学应用领域的研究重点。。

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. .