We study the mean curvature flow in 3dimensional 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 codimensiontwo mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer untrapped initial surface, a condition which resembles the meanconvexity 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 untrapped foliation asymptotically.
more » « less NSFPAR ID:
 10363349
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Communications in Mathematical Physics
 Volume:
 390
 Issue:
 3
 ISSN:
 00103616
 Page Range / eLocation ID:
 p. 11491173
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

null (Ed.)Abstract We study the phenomenon of TypeII curvature blowup in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate {(Tt)^{{(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higherdimensional analogue of the “Grim Reaper” solution for the curveshortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.more » « less

Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singularities can occur under the flow, what the flow looks like near these singularities, and what this implies for the structure of the singular set. At the end, we will briefly discuss how one may be able to use the flow in lowdimensional topology.

We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a nonminimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous minmax theory, developed for constant mean curvature hypersurfaces, can be extended to construct minmax prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that h has a sign.more » « less

Abstract We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Antide Sitter black hole initial data.more » « less

Abstract While it is well known from examples that no interesting “halfspace theorem” holds for properly immersed $n$dimensional selftranslating mean curvature flow solitons in Euclidean space $\mathbb{R}^{n+1}$, we show that they must all obey a general “bihalfspace theorem” (aka “wedge theorem”): two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori–Yau maximum principle. As an application we classify the closed convex hulls of all properly immersed (possibly with compact boundary) $n$dimensional mean curvature flow selftranslating solitons $\Sigma ^n$ in ${\mathbb{R}}^{n+1}$ up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman–Meeks in 1989 for minimal submanifolds: all of ${\mathbb{R}}^{n}$, halfspaces, slabs, hyperplanes, and convex compacts in ${\mathbb{R}}^{n}$.