Morse Theory, Harmonic Maps, and Poisson Structures

摘要奖：DMS-9971721主要研究人员：Kevin Corlette主要研究两个方向。第一个方向是利用Morse理论研究等变调和映射和其他变分问题。在某些情况下，存在年龄计量定义的变分问题的对称奇异解。在适当的情况下，可以利用Morse理论的技巧来推导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备的Hyperkahler流形的问题。Bando-Kobayashi和Tian-Yau的结果给出了拟投射簇上Ricci-Flat Kahler度量的存在性，实现了Fano簇中适当反非因子的互补。对这些指标来说，这似乎是一个非常严格的条件，即它们应该是Hyperkahler。然而，如果以除数的光滑性条件为轴，就知道存在非平凡的例子。我们建议寻找射影簇上的非平凡泊松结构，它可能与完全的超卡勒度规有关。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推动的。描述存在杨-米尔场的引力方程组的某些特解在过去的十年里引起了人们的极大关注，因为它们没有奇点，即没有黑洞。这组方程有一个无穷族的解，它们在极限上近似于一个黑洞解。提出的问题是，能否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，但还有一个类似的问题，涉及球面之间的调和映射(试图尽可能有效地将一个球面适配到另一个球面中)，它更容易理解。建议的第二部分涉及寻找几何空间的样本，称为Hyperkahler流形。在过去20年左右的时间里，出于一些原因，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，但也与经典和量子力学中的问题有关，并出现在弦理论的各种公式中。

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.