Morse Theory, Harmonic Maps, and Poisson Structures

莫尔斯理论、调和图和泊松结构

• 批准号: 9971721
• 负责人: Kevin Corlette
• 金额: $ 15.75万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971721&HistoricalAwards=false
• 关键词: Morse; Theory; Harmonic; Maps; Poisson

AbstractAward: DMS-9971721Principal Investigator: Kevin CorletteThe principal investigator proposes to work in two directions.The first is a study of equivariant harmonic maps and othervariational problems by means of Morse theory. There are certainsituations where a symmetric but singular solution of ageometrically defined variational problem exists. Under suitablecircumstances, techniques from Morse theory can be used to deducethe existence of additional nonsingular solutions of the sameproblem. The second direction is is related to the problem offinding complete hyperkahler manifolds. There are results ofBando-Kobayashi and Tian-Yau which give the existence ofRicci-flat Kahler metrics on quasiprojective varieties realizedas complements of suitable anticanonical divisors in Fanovarieties. It appears to be a very restrictive condition onthese metrics that they should be hyperkahler. However, if onerelaxes the smoothness conditions on the divisor, one knowsnontrivial examples exist. We propose a search for nontrivialPoisson structures on projective varieties which might beassociated with complete hyperkahler metrics.The work in the first part of the proposal was motivated by aquestion asked by a physicist who works on the theory of gravity.There are certain special solutions of the system of equationsdescribing gravity in the presence of a Yang-Mills field whichhave attracted a great deal of attention over the past decadebecause they have no singularities, i.e. no black holes. Thereis an infinite family of solutions to this system of equations,and they approximate a black hole solution in the limit. Thequestion asked was whether or not one can explain the existenceof these solutions based on the symmetry in the problem and the"shape" of the space of candidate solutions. This problem isvery difficult, but there is an analogous problem involvingharmonic maps between spheres (trying to fit one sphere insideanother as efficiently as possible) which is more accessible.The second part of the proposal involves an attempt to findexamples of geometric spaces called hyperkahler manifolds. Thesehave been objects of intense interest over the past 20 years orso, for a number of reasons. They arise in algebraic geometry,but also are related to issues in classical and quantummechanics, and have made an appearance in various formulations ofstring theory.

摘要奖项：DMS-9971721 首席研究员：Kevin Corlette 首席研究员建议在两个方向开展工作。第一个方向是利用莫尔斯理论研究等变调和映射和其他变分问题。 在某些情况下，几何定义的变分问题存在对称但奇异的解。 在适当的情况下，莫尔斯理论的技术可以用来推断同一问题的附加非奇异解的存在性。 第二个方向与寻找完整的超卡勒流形的问题有关。 Bando-Kobayashi 和 Tian-Yau 的结果给出了准射影簇上的 Ricci-flat Kahler 度量的存在性，该度量是作为 Fanovarieties 中合适的反规范除数的补充而实现的。 这些指标应该是 hyperkahler，这似乎是一个非常严格的条件。 然而，如果放宽除数的平滑条件，就知道存在不平凡的例子。 我们提议搜索射影簇上的非平凡泊松结构，这可能与完整的超卡勒度量相关。该提案第一部分的工作是由一位研究引力理论的物理学家提出的问题激发的。在存在杨-米尔斯场的情况下，描述引力的方程组有某些特殊的解，这些解在过去引起了极大的关注 十年因为它们没有奇点，即没有黑洞。 该方程组有无穷族解，它们在极限上近似于黑洞解。 提出的问题是，是否可以根据问题的对称性和候选解空间的“形状”来解释这些解的存在。 这个问题非常困难，但有一个类似的问题涉及球体之间的调和映射（试图尽可能有效地将一个球体拟合到另一个球体中），该问题更容易解决。该提案的第二部分涉及尝试找到称为超卡勒流形的几何空间的示例。 由于多种原因，在过去 20 年左右的时间里，这些物体一直引起人们的强烈兴趣。 它们出现在代数几何中，但也与经典和量子力学中的问题有关，并出现在弦理论的各种表述中。