Volume preserving curvature flows in Lorentzian manifolds

TL;DR

This paper studies volume-preserving curvature flows in Lorentzian manifolds, proving long-term existence, exponential convergence to constant curvature hypersurfaces, and analyzing stability and foliation properties.

Contribution

It establishes the long-time existence and convergence of curvature flows with volume preservation in Lorentzian manifolds, including various curvature functions.

Findings

Flows converge exponentially to constant F-curvature hypersurfaces

Long-term existence of the flows is proven under suitable conditions

Stability properties and foliations of constant F-curvature hypersurfaces are analyzed

Abstract

Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second elementary symmetric polynomial) and a volume preserving term. Under suitable assumptions we prove the long time existence of the flow and the exponential convergence of the corresponding graphs in the $C^\infty$-topology to a hypersurface of constant F-curvature. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.