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# Surveys in Differential Geometry

## Volume 12 (2007)

### Relativistic Teichmüller theory: a Hamilton-Jacobi approach to 2+1–dimensional Einstein gravity

Pages: 203 – 250

DOI: http://dx.doi.org/10.4310/SDG.2007.v12.n1.a6

#### Author

#### Abstract

We consider vacuum spacetimes in 2 + 1 dimensions defined on manifolds of the form M = Σ × **R** where Σ is a compact, orientable surface of genus > 1. By exploiting the harmonicity properties of the Gauss map for an arbitrary constant mean curvature (CMC) slice in such a spacetime we relate the Hamiltonian dynamics of the corresponding reduced Einstein equations to some fundamental results in the Teichmüller theory of harmonic maps. In particular we show, expanding upon an argument sketched by Puzio, that a global complete solution to the Hamilton-Jacobi equation for the reduced Einstein equations can be expressed in terms of the Dirichlet energy for harmonic maps defined over the surface Σ. While in principle this complete solution to the Hamilton-Jacobi equation determines all the solution curves to the reduced Einstein equations, one can derive a more explicit characterization of these curves through the solution of an associated (parametrized) Monge-Ampère equation. Using the latter we define a corresponding family of “ray structures” on the Teichmüller space of the chosen 2-manifold Σ. These ray structures are similar but complementary to a different family of such ray structures defined by M. Wolf and we herein derive a “relativistic interpretation” of both sets. We also use our Hamilton-Jacobi results to define complementary families of Lagrangian foliations of the cotangent bundle of the Teichmüller space of Σ and to provide the corresponding “relativistic interpretation” of the leaves of these foliations.