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# Communications in Analysis and Geometry

## Volume 13 (2005)

### Number 5

### Constant mean curvature foliations of simplicial flat spacetimes

Pages: 963 – 979

DOI: http://dx.doi.org/10.4310/CAG.2005.v13.n5.a6

#### Author

#### Abstract

Benedetti and Guadagnini have conjectured that the constant mean curvature foliation $M_\tau$ in a $2+1$ dimensional flat spacetime $V$ with compact hyperbolic Cauchy surfaces satisfies $\lim_{\tau \to -\infty} \ell_{M_\tau} = s_{\Tree}$, where $\ell_{M_\tau}$ and $s_{\Tree}$ denote the marked length spectrum of $M_\tau$ and the marked measure spectrum of the $\Re$-tree $\Tree$, dual to the measured foliation corresponding to the translational part of the holonomy of $V$, respectively. We prove that this is the case for $n+1$ dimensional, $n \geq 2$, simplicial flat spacetimes with compact hyperbolic Cauchy surface. A simplicial spacetime is obtained from the Lorentz cone over a hyperbolic manifold by deformations corresponding to a simple measured foliation.