Morse Theory, Harmonic Maps, and Poisson Structures

项目编号：dms -9971721首席研究员：Kevin corlett首席研究员提出了两个方向的工作。首先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，几何定义变分问题存在对称但奇异的解。在适当的情况下，可以利用莫尔斯理论的技术推导出同一问题的附加非奇异解的存在性。第二个方向是关于寻找完全超卡勒流形的问题。bando - kobayashi和Tian-Yau的结果给出了拟投影品种上的ricci -flat Kahler度量作为拟投影品种中合适的反正则因子的补的存在性。这似乎是一个非常严格的条件，对这些指标，他们应该是超kahler。然而，如果放宽除数上的平滑条件，我们知道存在非平凡的例子。我们提出了一个可能与完全超凯勒度量相关的射影变异上的非平凡泊松结构的搜索。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一个问题所激发的。在杨-米尔斯场存在的情况下，方程组有一些特殊的解，它们没有奇点，也就是说没有黑洞，因此在过去的十年里引起了人们的极大关注。这个方程组有无穷多个解，它们在极限下近似于黑洞的解。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，但是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装入另一个球体），这更容易理解。提案的第二部分涉及尝试寻找称为超卡勒流形的几何空间的例子。在过去的20多年里，由于一些原因，这些一直是人们非常感兴趣的对象。它们出现在代数几何中，但也与经典力学和量子力学中的问题有关，并出现在弦理论的各种表述中。

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.