Morse Theory, Harmonic Maps, and Poisson Structures

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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.