Nonlinear gravitons, null geodesics, and holomorphic disks

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 × 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 C P 3 with boundary on some totally real embedding of R P 3 into C 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