Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks

Details Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks

2005-04-28

arxiv.org/abs/math/0504582v1

Mathematics

Differential Geometry

56 pages, LaTeX2e

Scientific paper

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 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 K\"ahler metrics, and our methods give particularly sharp results in this more restrictive setting.

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