Mean curvature flow of graphs in generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition (Q2687964)

In this paper, the authors investigate the mean curvature flow for some special kind of spacelike hypersurfaces (namely graphs) in generalized Robertson-Walker (GRW) spacetimes, a special class of Lorentzian manifolds endowed with a closed conformal timelike vector field. They prove the following result: Let \(N^{n+1}=I\ltimes_fM^n\) be a GRW spacetime that obeys the null convergence condition and whose warping function \(f\) satisfies some specific conditions, and let \(\Omega\) be a compact convex domainin \(M\) and \(K\) its respective conformal cylinder. If \(\Sigma^n\) is a spacelike graph over \(\Omega\) that intersects \(K\) orthogonally, then there exists a longtime solution to the mean curvature flow problem whose metric is conformal to the one of the leaf \(M^n\) in asymptotic time. Furthermore, if \(\Sigma^n\) is mean convex, then the evolving graphs remain mean convex.