Mean Curvature Flow in Null Hypersurfaces and the Detection of MOTS

Abstract

We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.

Ricci flow on surfaces along the standard lightcone in the $3+1$-Minkowski spacetime

“…In the context of Lorentzian geometry, Hamilton's initial restriction to metrics of strictly positive scalar curvature translates to a physically reasonable assumption on the representing codimension-2 surface in the Minkowski spacetime. From the Gauß equation, we are moreover able to conclude the equivalence of Ricci flow and null mean curvature flow along the lightcone first studied by Roesch-Scheuer [17]. As defined by Roesch-Scheuer, null mean curvature flow…”

“…as first studied by Roesch-Scheuer [17]. Note that since L(r) = 1, the above is equivalent to the following evolution equation for ω…”

“…For the sake of simplicity, we will keep the discussion of null geometry to the standard Minkowski lightcone. For the interested reader, we refer to [16,17] for a more complete and general introduction to null geometry. For our purpose, it is most convenient to introduce the null coordinates v := r + t, u := r − t on R 3,1 .…”

Ricci flow on surfaces along the standard lightcone in the $$3+1$$-Minkowski spacetime

“…Note that null mean curvature flow is the projection of codimension-2 mean curvature flow onto N in direction of the null generator L, and that the flow is at least locally always equivalent to a scalar parabolic equation, cf. [18].…”

“…Under some mild assumptions on the choice of null generator L and assuming that appropriate barriers exists, Roesch-Scheuer show that the flow exists for all times and converges smoothly to a MOTS, cf. [18,Theorem 1.1]. In particular, their assumptions are satisfied for the lightcone in the Schwarzschild spacetime (of positive mass).…”