A Twistor Approach To The Einstein Metric On K3 (calabi Conjecture, Surface, Yau's Theorem, Ricci Curvature).

Abstract

An important consequence of Yau's solution to Calabi's conjecture is the existence of a Kahler-Einstein metric on the complex surface K3. However, Yau's existence proof for the resulting P.D.E.'s, using Schauder estimates and the continuity method, is only implicit, and does not appear amenable to make explicit calculations or asymptotics. The thrust of this thesis is to give an entirely different proof of the existence of the Kahler-Einstein metric on some K3, using the geometrical methods of twistor theory and complex deformation theory. Although our proof is still globally implicit, there is explicit local information built-in in the following sense: the twistor space for K3 is the complex 3-manifold fibering over K3 with P(,1) fibers whose complex structure actually encodes the K.-E. metric on K3. When this fibering is restricted to a neighborhood of a nodal curve (in a Kummer model of K3), the twistor space is biholomorphic to the one for another metric--the Eguchi-Hanson metric. This metric is explicitly known, and our proof shows essentially that the K3 metric approaches the Eguchi-Hanson metric in these regions. This qualitative behavior of the K3 metric was conjectured by the physicist Don Page, who gave a physical picture of the free parameters of this metric in these terms, and showed also the link with S. Hawking's count in terms of ((+OR-)) self-dual harmonic 2-forms. By using relative complex deformation theory on the twistor space, we put all these parameter counts on a firm footing, reinterpreting them in terms of complex structures.