Nonlinear gravitons, null geodesics, and holomorphic disks

Abstract

We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature $(++--)$ with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on $S^2\times S^2$, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the surprising global property that its null geodesics are all periodic. Each such conformal structure arises from a family of holomorphic disks in $\mathbb{C}{\mathbb{P}}_3$ with boundary on some totally real embedding of $\mathbb{R}{\mathbb{P}}^3$ into $\mathbb{C}{\mathbb{P}}_3$. Some of these conformal classes are represented by scalar-flat indefinite Kähler metrics, and our methods give particularly sharp results in connection with this special case