Graduate Thesis Defenses 2008
Geometric Quantization and Dynamical Constructions on the Space of Kähler Metrics
Friday, February 15, 2008
2:00 pm, Room 2-131
Committee: Gang Tian, Princeton (thesis advisor), Tomasz Mrowka, Victor Guillemin
This Thesis is concerned with the study of the geometry and structure of the space of Kahler metrics representing a fixed cohomology class on a compact Kahler manifold. The first part of the Thesis is concerned with a problem of geometric quantization: Can the geometry of the infinite-dimensional space of Kahler metrics be approximated in terms of the geometry of the finite-dimensional spaces of Fubini-Study Bergman metrics sitting inside it? We restrict to toric varieties and prove the following result: Given a compact Riemannian manifold with boundary and a smooth map from its boundary into the space of toric Kahler metrics there exists a harmonic map from the manifold with these boundary values and, up to the first two derivatives, it is the limit of harmonic maps from the Riemannian manifold into the spaces of Bergman metrics. This generalizes previous work of Song-Zelditch on geodesics in the space of toric Kahler metrics. In the second part of the Thesis we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamical systems on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow as well as another flow on the space of Kahler metrics. We introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics that arise as discretizations of these flows. As an application, we address several questions in Kahler geometry related to canonical metrics, energy functionals, the Moser-Trudinger-Onofri inequality, Nadel-type multiplier ideal sheaves, and the structure of the space of Kahler metrics.