Principal Investigator: Jian Song
This proposal concerns existence and regularity problems of the nonlinear Monge-Ampere type equations from geometry and physics. The problem of finding canonical metrics on a compact Kahler manifold has been the subject of intense study over the last few decades. In his solution to Calabi's conjecture, Yau proved the existence of a Kahler-Einstein metric on compact Kahler manifolds with vanishing or negative first Chern class. An alternative proof of Calabi's conjecture is given by Cao using the Kahler-Ricci flow. However, most projective algebraic varieties do not have definite or trivial first Chern classes. Recently Perelman has made major breakthrough in Hamilton's program as an approach to the Poincare conjecture and Thurston's geometrization conjecture. The principal investigator proposes to study the canonical metrics on the canonical models of projective varieties of positive Kodaira dimension by applying the Kahler-Ricci flow. Such canonical metrics are constructed by the deformation of the Kaher-Ricci flow on minimal projective surfaces of positive Kodaira dimension. These generalized Kahler-Einstein metrics can be considered as an analytic version of the abundance conjecture in algebraic geometry and will also lead to new advances in the understanding and application of the Ricci flow. In Donaldson's far reaching program, the geometry of the infinite dimensional symmetric space of Kahler metrics in a fixed class is related to the existence and uniqueness of constant scalar curvature Kahler metrics. The principal investigator also intends to study the uniform approximation problem of the Monge-Ampere geodesics in infinite dimensional symmetric space by those in the finite dimensional Bergman spaces on toric varieties. The precise understanding of this problem will give new insight into the conjecture proposed by Yau between the relation of constant scalar curvature Kahler metrics and certain stability in the sense of geometric invariant theory. The principal investigator will also apply the moment map point of view and study various geometric flows arising from Kahler geometry as well as symplectic geometry proposed by Donaldson. The first is the J-flow, which is the gradient flow of functional related to the Mabuchi energy. The second is a moment map flow in a hyperkahler four manifold. The principal investigator intends to study the question of convergence and singularities os such parabolic flows.
Since the discovery of the general relativity, geometric analysis has become crucial to both mathematicians and physicists. Problems in this proposal arise naturally from our attempts to understand nonlinear differential equations from geometry and physics. The solutions to these problems will contribute to various fields of sciences such as physics and cosmology in the deep understanding of our universe. The method of analyzing the singularities of nonlinear equations will have wide applications in engineering and economics.