Jan Metzger

The Area of Horizons and the Trapped Region

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers, curvature estimates, together with a novel surgery construction for marginal surfaces. These results are applied to characterize the boundary of the trapped region.

Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivity of the isoperimetric mass established by G. Huisken we prove an existence result for large isoperimetric regions in general asymptotically flat three manifolds with non-negative scalar curvature.

The Time evolution of marginally trapped surfaces

In previous work, we have shown the existence of a dynamical horizon or marginally outer trapped tube (MOTT) containing a given strictly stable marginally outer trapped surface (MOTS). In this paper, we show some results on the global behavior of MOTTs assuming the null energy condition. In particular, we show that MOTSs persist in the sense that every Cauchy surface in the future of a given Cauchy surface containing a MOTS also must contain a MOTS. We describe a situation where the evolving outermost MOTS must jump during the coalescence of two separate MOTSs. We furthermore characterize the behavior of MOTSs in the case that the principal eigenvalue vanishes under a genericity assumption. This leads to a regularity result for the tube of outermost MOTSs under the genericity assumption. This tube is then smooth up to finitely many jump times. Finally we discuss the relation of MOTSs to singularities of a spacetime.

Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces

The aim of this paper is to accurately describe the blowup of Jang’s equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponential convergence with an identifiable rate near a strictly stable MOTS.

Minimizers of the Willmore functional with a small area constraint

Abstract We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

No mass drop for mean curvature flow of mean convex hypersurfaces

A possible evolution of a compact hypersurface in R by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow. As a further application of the techniques used above we give a new variational formulation for mean curvature flow of mean convex hypersurfaces.

Numerical computation of constant mean curvature surfaces using finite elements

This paper presents a method for computing two-dimensional constant mean curvature surfaces. The method in question uses the variational aspect of the problem to implement an efficient algorithm. In principle, it is a flow-like method in that it is linked to the gradient flow for the area functional, which gives reliable convergence properties. In the background, a preconditioned conjugate gradient method works, which gives the speed of a direct elliptic multigrid method.

The merger of small and large black holes

We present simulations of binary black-hole mergers in which, after the common outer horizon has formed, the marginally outer trapped surfaces (MOTSs) corresponding to the individual black holes continue to approach and eventually penetrate each other. This has very interesting consequences according to recent results in the theory of MOTSs. Uniqueness and stability theorems imply that two MOTSs which touch with a common outer normal must be identical. This suggests a possible dramatic consequence of the collision between a small and large black hole. If the penetration were to continue to completion, then the two MOTSs would have to coalesce, by some combination of the small one growing and the big one shrinking. Here we explore the relationship between theory and numerical simulations, in which a small black hole has halfway penetrated a large one.

Surfaces with maximal constant mean curvature

In this chapter we shall review some basic aspects of the theory of surfaces with constant mean curvature. Surfaces with constant mean curvature will arise as solutions of a variational problem associated to the area functional and related with the classical isoperimetric problem. We state the first and second variation formula for the area and we give the notion of stability of a cmc surface. Next, we introduce the complex analysis as a basic tool in the theory and this will allow to prove the Hopf theorem. Also, we compute the Laplacians of some functions that contain geometric information of a cmc surface. As the Laplacian operator is elliptic, the maximum principle yields height estimates of a graph of constant mean curvature. Finally, and with the aid of the expression of these Laplacians, we will derive the Barbosa-do Carmo theorem that characterizes a round sphere in the family of closed stable cmc surfaces of Euclidean space.

Mini-Workshop: The Willmore Functional and the Willmore Conjecture

The Willmore functional evaluated on a surface immersed into Euclidean space is given by the L-norm of its mean curvature. The interest for studying this functional comes from various directions. First, it arises in applications from biology and physics, where it is used to model surface tension in the Helfrich model for bilipid layers, or in General Relativity where it appears in Hawking’s quasi-local mass. Second, the mathematical properties justify consideration of the Willmore functional in its own right. The Willmore functional is one of the most natural extrinsic curvature functionals for immersions. Its critical points solve a fourth order Euler-Lagrange equation, which has all minimal surfaces as solutions. Mathematics Subject Classification (2010): 53C42, 58E12, 35J48, 35J60, 49Q10. Introduction by the Organisers In recent years there has been substantial progress concerning analytical and geometrical questions related to the Willmore functional. Highlights include the study of surfaces with square integrable second fundamental form, the compactness of W -conformal immersions, the regularity of weak solutions of the Willmore equation and the resolution of the longstanding Willmore conjecture. The aim of this mini-workshop was to bring together people involved in propelling the above mentioned research highlights. In particular, our intention was to make a connection between the experts on minimal surfaces and corresponding min-max techniques that were a crucial ingredient in the proof of the Willmore conjecture, and the experts for the analysis developed for second order curvature functionals such as the Willmore functional. 2120 Oberwolfach Report 37/2013 For this purpose two mini-courses were delivered by Fernando Marques (Minmax theory and the Willmore conjecture) and Tristan Riviere (The variations of the Willmore Lagrangian, a parametric approach). Moreover, every participant gave a talk, with plenty of time left for discussions. Mini-Workshop: The Willmore Functional and the Willmore Conjecture 2121 Mini-Workshop: The Willmore Functional and the Willmore Conjecture