Partial Differential Equations on Complex and Symplectic Manifolds

Columbia University, New York NY

Investigators

Abstract

The PI proposed research focuses on several basic problems related to the geometry of complex and symplectic manifolds, which can be studied using nonlinear PDEs. In the first project the PI will study a recent conjecture of Donaldson that aims at extending Yau's theorem in Kahler geometry to symplectic four-manifolds, building on his work with Weinkove and Yau. If proved, this conjecture would provide a powerful new tool to construct symplectic forms on compact symplectic four-manifolds, and would have striking applications to symplectic topology. The second project regards the geometry of compact Calabi-Yau manifolds, and specifically the way in which Ricci-flat Kahler metrics on a Calabi-Yau manifold can degenerate when their cohomology class approaches the boundary of the Kahler cone. These degenerations have also been studied by string theorists in connection with mirror symmetry. The PI proposes to continue his study of these degenerations, as well as investigating Ricci-flat metrics on a family of quintic threefolds near a large complex structure limit. The third project falls in the area of canonical metrics on compact Kahler manifolds, such as Kahler-Einstein or constant scalar curvature Kahler metrics. It is believed that the existence of such canonical metrics should be equivalent to the algebraic stability of the manifold. The PI will study this using two natural evolution equations associated to these problems, the Kahler-Ricci flow and the Calabi flow, with the aim of connecting the limiting behaviour of the flows to algebraic stability through the use of natural energy functionals. The final project also involves canonical Kahler metrics, and more specifically the problem of existence of constant scalar curvature Kahler metrics on complex surfaces with ample canonical bundle in cohomology classes that are known to be stable. Most of the problems that we will consider, for example the Einstein equations, were originally discovered by physicists who were searching for models of the fundamental laws of nature. More recently, geometric aspects closely related to the proposed research have found applications in high energy physics, and are being used to deepen our understanding of the Universe and of elementary particles. The geometric ideas of the PI's research revolve around the problem of finding the optimal shape of a geometric space, the one with the largest possible symmetry, and understanding the possible singularities that form in spaces where such an optimal shape does not exist. Any progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences.