Problems in Geometric Analysis: Harmonic Maps and Holomorphic Vector Bundles

几何分析中的问题：调和映射和全纯向量丛

• 批准号: 0505512
• 负责人: Richard Wentworth
• 金额: $ 24.02万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505512&HistoricalAwards=false
• 关键词: Problems; Geometric; Analysis; Harmonic; Maps

The first project deals with energy minimizing maps to the Weil-Petersson completion of Teichmueller space. The proposed research will establish sufficient regularity for higher dimensional domains so that harmonic map theory may be used to answer rigidity questions. The second part of the proposal continues work on the Yang-Mills flow on higher dimensional Kaehler manifolds. A special focus is a comparison of the analytic singularities which occur along the flow with the algebraic singularities associated to Harder-Narasimhan filtrations of holomorphic vector bundles. The third project consists of topics related to representation varieties of surface groups. Included is a study of properties of the energy functional on Teichmueller space defined by harmonic maps associated to surface group representations. Results in this direction will have implications for the dynamics of the mapping class group action on the moduli space of representations.A significant branch of mathematical inquiry has been the relationship between the geometric, analytic, and algebraic properties of manifolds. Manifolds are higher dimensional generalizations of curves and surfaces, and they appear in a variety of situations in pure and applied mathematics. The research projects in this proposal will further our understanding of some of these objects. The equations studied -- energy minimizing maps and the Yang-Mills flow -- have their origins in the mathematical description of the physical world. They are therefore of great importance to both mathematicians and physicists.

第一个项目涉及能量最小化映射到Teichmueller空间的Weil-Petersson完成。所提出的研究将为高维域建立足够的正则性，以便调和映射理论可以用来回答刚性问题。第二部分的建议继续工作的杨米尔斯流高维Kaehler流形。 一个特别的重点是比较的解析奇点发生沿着流与代数奇点相关的Harder-Narasimhan过滤的全纯向量丛。第三个项目包括与曲面群的各种表示相关的主题。 包括一个研究的性质的能量功能Teichmueller空间定义的调和映射与表面群表示。 这一方向的结果将对表示的模空间上的映射类群作用的动力学产生影响。数学研究的一个重要分支是流形的几何、解析和代数性质之间的关系。流形是曲线和曲面的高维推广，它们出现在纯数学和应用数学中的各种情况下。本提案中的研究项目将进一步加深我们对其中一些物体的理解。所研究的方程--能量最小化映射和杨-米尔斯流--起源于物理世界的数学描述。 因此，它们对数学家和物理学家都非常重要。