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

Abstract

Award: DMS-0604805

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.