Volume preserving curvature flows in Lorentzian manifolds

Abstract

Let $N$ be a $(n+1)$-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface $\mathcal{S}_0$ and $F$ a curvature function, either the mean curvature $H$, the root of the second symmetric polynomial $\si_2 = \sqrt{H_2}$ or a curvature function of class $(K^*)$, a class of curvature functions which includes the $n$-th root of the Gaussian curvature $\si_n = K^{\frac{1}{n}}$. We consider curvature flows with curvature function $F$ and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the $C^\infty$-topology to a hypersurface of constant $F$-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant $F$-curvature hypersurfaces.