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