Morse Theory, Harmonic Maps, and Poisson Structures

AbstractAward：DMS-9971721首席研究员：Kevin Corlette首席研究员提出了两个方向的工作，第一个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，利用莫尔斯理论的技巧，可以推导出同一问题的附加非奇异解的存在性. 第二个方向是关于完备超kahler流形的寻找问题。 Bando-Kobayashi和Tian-Yau的结果给出了拟投射簇上Ricci-平坦Kahler度量的存在性，这些度量是Fano簇中合适的反正则因子的补。 这似乎是一个非常严格的条件对这些指标，他们应该hyperkahler。 然而，如果放松除数的光滑条件，就知道存在非平凡的例子。 本文提出了一个在射影簇上寻找可能与完备超kahler度量有关的非平凡Poisson结构的方法.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.在存在杨-米尔斯场在过去的十年里吸引了大量的关注，因为它们没有奇点，也就是说没有黑洞。 这个方程组有无穷多个解，它们在极限下近似于黑洞解。 所问的问题是，是否可以解释这些解决方案的对称性的基础上存在的问题和“形状”的候选解决方案的空间。 这个问题是非常困难的，但有一个类似的问题涉及到球之间的调和映射（试图尽可能有效地将一个球装入另一个球），这是更容易理解的。该建议的第二部分涉及到试图找到一个称为超卡勒流形的几何空间的例子。 在过去的20年里，这些都是人们强烈兴趣的对象，原因有很多。 它们出现在代数几何中，但也与经典力学和量子力学的问题有关，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.