Nonlinear Geo metric Equations of Monge-Ampere Type and Canonical Metrics

点击翻译按钮获取中文摘要

AbstractAward: DMS-0604805Principal Investigator: Jian SongThis proposal concerns existence and regularity problems of thenonlinear Monge-Ampere type equations from geometry and physics.The problem of finding canonical metrics on a compact Kahlermanifold has been the subject of intense study over the last fewdecades. In his solution to Calabi's conjecture, Yau proved theexistence of a Kahler-Einstein metric on compact Kahler manifoldswith vanishing or negative first Chern class. An alternativeproof of Calabi's conjecture is given by Cao using theKahler-Ricci flow. However, most projective algebraic varietiesdo not have definite or trivial first Chern classes. RecentlyPerelman has made major breakthrough in Hamilton's program as anapproach to the Poincare conjecture and Thurston's geometrizationconjecture. The principal investigator proposes to study thecanonical metrics on the canonical models of projective varietiesof positive Kodaira dimension by applying the Kahler-Ricciflow. Such canonical metrics are constructed by the deformationof the Kaher-Ricci flow on minimal projective surfaces ofpositive Kodaira dimension. These generalized Kahler-Einsteinmetrics can be considered as an analytic version of the abundanceconjecture in algebraic geometry and will also lead to newadvances in the understanding and application of the Ricciflow. In Donaldson's far reaching program, the geometry of theinfinite dimensional symmetric space of Kahler metrics in a fixedclass is related to the existence and uniqueness of constantscalar curvature Kahler metrics. The principal investigator alsointends to study the uniform approximation problem of theMonge-Ampere geodesics in infinite dimensional symmetric space bythose in the finite dimensional Bergman spaces on toricvarieties. The precise understanding of this problem will givenew insight into the conjecture proposed by Yau between therelation of constant scalar curvature Kahler metrics and certainstability in the sense of geometric invariant theory. Theprincipal investigator will also apply the moment map point ofview and study various geometric flows arising from Kahlergeometry as well as symplectic geometry proposed byDonaldson. The first is the J-flow, which is the gradient flow offunctional related to the Mabuchi energy. The second is a momentmap flow in a hyperkahler four manifold. The principalinvestigator intends to study the question of convergence andsingularities os such parabolic flows.Since the discovery of the general relativity, geometric analysishas become crucial to both mathematicians and physicists.Problems in this proposal arise naturally from our attempts tounderstand nonlinear differential equations from geometry andphysics. The solutions to these problems will contribute tovarious fields of sciences such as physics and cosmology in thedeep understanding of our universe. The method of analyzing thesingularities of nonlinear equations will have wide applicationsin engineering and economics.