Volume preserving curvature flows in Lorentzian manifolds (Q1937821)

The author considers an \((n+1)\)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface \(S_0\) and a curvature function \(F\), either the mean curvature \(H\), the root of the second symmetric polynomial \(\sigma_2= \sqrt{H_2}\) or a curvature function of class \((K^*)\), a class of curvature functions which includes the \(n\)th root of the Gaussian curvature \(\sigma_n= K^{{1\over n}}\). He considers curvature flows with curvature function \(F\) and a volume preserving term and proves 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, the author examines stability properties and foliations of constant \(F\)-curvature hypersurfaces.