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 S2 × S2, there is an infinitedimensional 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 CP3 with boundary on some totally real embedding of RP3 into CP3. An interesting sub-class of these conformal structures are represented by scalar-flat indefinite Kähler metrics, and our methods give particularly sharp results in this more restrictive setting.