Mean Curvature Flow in Null Hypersurfaces and the Detection of MOTS

Henri Roesch,Julian Scheuer

doi:10.1007/s00220-022-04326-9

Henri Roesch, Julian Scheuer

Open Access

https://doi.org/10.1007/s00220-022-04326-9

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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.

Highlights

Let (M, g) be a four dimensional, time-oriented Lorentzian manifold or spacetime with Levi-Civita connection D, where for convenience we write X, Y := g(X, Y )for vector fields X, Y of M
In this paper we propose a new method to find marginally outer trapped surface (MOTS), namely by employing the mean curvature flow (MCF) in a null hypersurface of a spacetime
Mean Curvature Flow in Null Hypersurfaces a smooth manifold into a Riemannian or Lorentzian ambient space, MCF can concisely be written as x = x = H, where is the Laplace-Beltrami operator with respect to the metric induced by x(t, ·) and x = x(·, ξ ) is the velocity of the curve t → x(t, ξ ), ξ being an element of the embeddings’ common domain, which will be S2 in this paper

Summary

Introduction

Let (M, g) be a four dimensional, time-oriented Lorentzian manifold or spacetime with Levi-Civita connection D, where for convenience we write. Mean Curvature Flow in Null Hypersurfaces a smooth manifold into a Riemannian or Lorentzian ambient space, MCF can concisely be written as x = x = H , where is the Laplace-Beltrami operator with respect to the metric induced by x(t, ·) and x = x(·, ξ ) is the velocity of the curve t → x(t, ξ ), ξ being an element of the embeddings’ common domain, which will be S2 in this paper. In this paper we show that (1.1) is capable of doing the following: Given a null hypersurface N supporting a trapped surface or MOTS, we identify fairly generic constraints on N for which our mean curvature flow (1.1) from any outer un-trapped initial cross-section exists for all times and converges smoothly to a MOTS. Under some sufficient conditions on a null hypersurface N , we prove the long-time existence of the mean curvature flow for spacelike spherical cross-sections within N and show that it converges to a MOTS. We may assume that 0 < T ∗ ≤ ∞ is the maximal time of smooth spacelike existence for both equations (Fig. 1)

Estimates

Null Cones

Existence of a Global Outer Un-trapped Foliation