We consider volume preserving curvature flows in globally hyperbolic Lorentzian manifolds with a compact Cauchy hypersurface. Possible choices of curvature functions F include the mean curvature, the square root of the second elementary symmetric polynomial as well as the n-th root of the Gaussian curvature.
Under the assumption of barriers and some curvature assumptions on the ambient manifold we prove long time existence of the flow and exponential convergence to a hypersurface of constant F-curvature. Furthermore we examine stability properties and foliations of the ambient manifold by constant
F-curvature hypersurfaces.
This paper has appeared in "Calc. Var. and Partial Differential Equations". Springer-Verlag is the copyright holder of this article. The original publication is available on http://www.springerlink.com/content/41624268468x07h3/.