2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75037We 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 x 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 CP_3 with boundary on some totally real embedding of RP^3 into CP_3. 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.56 pages, LaTeX2eDifferential GeometryGeneral Relativity and Quantum CosmologyHigh Energy Physics - TheoryMathematical PhysicsComplex Variables53C50; 32L25;32G10; 14J30text