## Duke Mathematical Journal

### 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

#### Article information

Source
Duke Math. J., Volume 136, Number 2 (2007), 205-273.

Dates
First available in Project Euclid: 21 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1166711369

Digital Object Identifier
doi:10.1215/S0012-7094-07-13621-4

Mathematical Reviews number (MathSciNet)
MR2286630

Zentralblatt MATH identifier
1113.53032

#### Citation

Lebrun, Claude; Mason, L. J. Nonlinear gravitons, null geodesics, and holomorphic disks. Duke Math. J. 136 (2007), no. 2, 205--273. doi:10.1215/S0012-7094-07-13621-4. https://projecteuclid.org/euclid.dmj/1166711369

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