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 \({{\sigma}_{2}=\sqrt{H_{2}}}\) or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature \({{\sigma}_{n}= K^{\frac{1}{n}}}\). 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 ∞-topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.
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Communicated by G. Huisken.
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Makowski, M. Volume preserving curvature flows in Lorentzian manifolds. Calc. Var. 46, 213–252 (2013). https://doi.org/10.1007/s00526-011-0481-0
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DOI: https://doi.org/10.1007/s00526-011-0481-0