The Erwin Schroedinger Institute for Mathematical Physics

Conference: Between Geometry and Relativity


Alessandro Carlotto (ETH Zurich)

Title: From scalar curvature rigidity phenomena to min-max embedded geodesic lines

Abstract: In recent years, there have been significant advances in our understanding of the large-scale geometry of asymptotically flat 3-manifolds of non-negative scalar curvature. Among other results, positivity of the ADM mass has been proven to be incompatible with the existence of complete (non-compact) area-minimizing surfaces (for asymptotically flat data) and even with the existence of stable minimal surfaces (for asymptotically Schwarzschildean data). At a geometric level, one is then naturally led to the well-known question concerning existence of unstable but asymptotically planar minimal surfaces. In ambient dimension two (namely: in the context of initial data for 1+2 spacetimes) this connects with the classical problem posed by Cohn-Vossen in 1936 about properly embedded geodesic lines on complete Riemannian manifolds. This is one of the most fundamental global problems in Differential Geometry and yet even in the simplest case of complete Riemannian metrics on the plane a general existence result was only obtained in 1981 by V. Bangert. But can we say more when we restrict to asymptotically flat surfaces of non-negative scalar curvature, namely in the physical scenario described above? In this talk, I shall present some recent joint work with Camillo De Lellis where we employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, where minimization schemes are inevitably doomed to fail, with full control on the Morse index and the asymptotic behaviour of the lines themselves.

Mattias Dahl (Stockholm)

Title: Outermost apparent horizons with non-trivial topology.

Abstract: An outermost apparent horizon in an asymptotically Euclidean manifold is a closed minimal hypersurface enclosing all other such hypersurfaces. In three dimensions, Hawking's black hole topology theorem states that an outermost apparent horizon must have a metric with positive scalar curvature if the surrounding manifold has non-negative scalar curvature. This has been generalized to higher dimensions by Cai, Galloway and Schoen.

It is conceivable that the existence of a positive scalar curvature metric is the only restriction on a bounding manifold to be an outermost apparent horizon. In this talk I want to describe some new metrics with outermost apparent horizons having non-trivial topology. These examples are constructed by a conformal blow up along a submanifold in Euclidean space, and the horizons are tubes around this submanifold. I will also discuss possible ways to go further and prove that all bounding manifolds with positive scalar curvature can be outermost horizons.

The results presented are joint work with Eric Larsson.

Michael Eichmair (Vienna)

Title: The large-scale isoperimetric structure of initial data

Abstract: A small geodesic ball with center at a point of positive scalar curvature encloses more volume with its surface area than would be possible in Euclidean space. This classical observation about scalar curvature in the small has a remarkable large-scale analogue obtained in recent joint work with O. Chodosh, Y. Shi, and H. Yu: Let (M, g) be an asymptotically flat Riemannian three-manifold with non-negative scalar curvature and which is not flat. For every V>0 sufficiently large, among all regions in M of volume V, there is a unique one whose boundary area is least. The boundaries ∑V of these isoperimetric regions foliate the complement of a compact subset of M, and

mADM = limV→∞ 2/area(∑V) (V - area(∑V)3/2/(6π1/2) ).

Helmut Friedrich (Golm)

Title: Einstein-lambda-matter flows near future time-like infinity

Abstract: In four space-time dimensions we consider future complete solutions to Einstein's field equations with positive cosmological constant and various matter fields and discuss their asymptotic behaviour at future time-like infinity.

Greg Galloway (Miami)

Title: Topology and singularities in general relativity

Abstract: A theme of long standing interest to the speaker is the relationship between the topology of spacetime and the occurrence of singularities. We will discuss several results exemplifying this relationship (some older, some newer) in both the asymptotically flat and cosmological settings. The proofs involve a number of interesting geometric elements, e.g., existence results for minimal surfaces and marginally outer trapped surfaces, and developments in low dimensional topology related to geometrization. The results to be discussed involve, in particular, work with a number of collaborators: Lars Andersson, Mattias Dahl, Michael Eichmair, Eric Ling, and Dan Pollack.

Lan-Hsuan Huang (UConn)

Title: Equality in the Spacetime Positive Mass Theorem

Abstract: We will discuss a variational proof to the equality of the spacetime positive mass theorem, based on a joint work with Dan Lee. The theorem says that for an asymptotically flat initial data set with the dominant energy condition, if the ADM energy-momentum vector is null, then the vector must be zero. Previously the result was only known for spin manifolds due to R. Beig and P. Chrusciel in three dimensions and to P. Chrusciel and D. Maerten in higher dimensions. Our proof removes the spin assumption. The central idea is to consider a modified Regge-Teitelboim Hamiltonian, used the modified constraint operator introduced in a joint work with Justin Corvino. We will also discuss other applications of the modified constraint operator for localized deformations of the dominant energy condition.

Gerhard Huisken (Tübingen)

Title: Inverse mean curvature flow for entire graphs

Abstract: The lecture describes joint work with P. Daskalopoulos on a priori estimates and the existence of longtime solutions to inverse mean curvature flow in the non-compact setting.

Nicolaos Kapouleas (Providence)

Title: Recent gluing constructions in Differential Geometry

In a large part of my talk I will concentrate on Doubling constructions for minimal surfaces by PDE methods and in particular singular perturbation methods. I will first introduce doubling constructions and discuss an early construction doubling the Clifford torus in collaboration with Yang and (briefly) extensions of this by David Wiygul. I will then discuss the Linearized Doubling (LD) approach (see a paper by the speaker to appear in JDG, and a preprint and ongoing work in collaboration with Peter McGrath), and its applications on doublings of the equatorial two-sphere in the round three-sphere and (ongoing work) for other minimal surfaces. I will then discuss work with Brendle (to appear in CAPM) on a question of Page on Ricci-flat four-manifolds and a related construction of ancient solutions of the Ricci flow. Finally depending on time left I will briefly discuss desingularizing constructions for minimal surfaces and other geometric gluing constructions.

Michael Kunzinger (Vienna)

Title: Singularity theorems in regularity C1,1

Abstract: The singularity theorems of General Relativity, initiated by R. Penrose in 1965 and extended by S. Hawking, G.F.R. Ellis, R. Geroch, G. Galloway, J. Senovilla, and many others, constitute a major milestone in the understanding of solutions to the Einstein equations. They predict the existence of singularities (incomplete causal geodesics) for spacetimes that satisfy physically reasonable assumptions. A shortcoming of the theorems is that they typically do not make statements on the character of the singularities they predict, e.g. whether curvature blows up at a singularity. In principle, a spacetime might be singular merely due to the differentiability of the spacetime metric dropping below C2, with discontinuous but bounded curvature. Such a drop in the regularity from C2 to C1,1 would correspond, via the field equations, to a finite jump in the matter variables, a situation that could hardly be regarded as singular from the viewpoint of physics. Also for the singularity theorems themselves the natural differentiability class is C1,1, as this is the minimal condition that ensures unique solvability of the geodesic equations. Recent progress in low-regularity Lorentzian geometry has allowed to show that, in fact, the classical singularity theorems of Penrose, Hawking, and Hawking–Penrose remain valid for C1,1-metrics. We will report on these results and the methods from low regularity causality and comparison geometry that were employed in the proofs. This is joint work with M. Graf, J.D. Grant, M. Stojkovic, R. Steinbauer, and J.A. Vickers.

Marc Mars (Salamanca)

Title: Geometry of null shells

Abstract: After reviewing the abstract hypersurface data formalism, I will concentrate on the case of null hypersurfaces. The abstract notion of shell will be introduced and its relation with the more usual concept that arises in the matching theory will be discussed. Then I will write down the fundamental equations that describe null shells in this formalism and analyze the jumps across the shell of various geometric quantities, including the Hawking energy and the total Bondi energy. Applications to the so-called shell Penrose inequality in the Minkowski and other background spacetimes will also be presented.

Jan Metzger (Potsdam)

Title: On the uniqueness of small surfaces minimizing the Willmore functional subject to a small area constraint

Abstract: We consider the Willmore functional for surfaces immersed in a compact Riemannian manifold M and study minimizers subject to a small area constratint. We show that if the scalar curvature of M has a non-degenerate maximum then for small enough area these minimizers are unique. This is joint work with Tobias Lamm and Felix Schulze.

Pengzi Miao (Miami)

Title: Minimal hypersurfaces, isometric embeddings, and manifolds with nonnegative scalar curvature

Abstract: Compact manifolds with nonnegative scalar curvature with boundary is a basic subject of study in the quasi-local mass problem in relativity. If the boundary of the manifold is isometric to a strictly convex hypersurface in the Euclidean space, a fundamental result of Shi and Tam asserts that the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In 3-dimension, Shi-Tam's result is equivalent to the Riemannian positive mass theorem.

In this talk, we present a supplement to Shi-Tam's theorem by considering manifolds which can be viewed as bodies surrounding horizons in an initial data set. More precisely, we consider a compact manifold with nonnegative scalar curvature, whose boundary consists of two parts, the outer boundary and the horizon boundary. Here the horizon boundary is the union of all closed minimal hypersurfaces in the manifold and the outer boundary is assumed to be a topological sphere. By assuming the outer boundary is isometric to a suitable 2-convex hypersurface in a Schwarzschild manifold of positive mass m, we establish an inequality relating m, the area of the horizon boundary, and two weighted total mean curvatures of the outer boundary and the hypersurface in the Schwarzschild manifold. In 3-dimension, this inequality is equivalent to the Riemannian Penrose inequality. This talk is based on joint work with Siyuan Lu.

André Neves (Chicago)

Title: Weyl Law and the volume spectrum

Abstract: The volume spectrum was introduced by Gromov in the 70’s. Recently, with Liokumovich and Marques, we proved a Weyl Law for the volume spectrum that was conjectured by Gromov. I will talk about how a better understanding of the volume spectrum would help answering some open questions for minimal surfaces or volume of nodal sets.

Richard Schoen (Irvine)

Title: The high dimensional positive energy theorem

Abstract: We will discuss the problem of proving the positive energy theorem in cases not directly covered by the Dirac operator approach or the minimal hypersurface approach. These are the cases when the dimension is greater than 8 and the manifold is not spin. The proof is accomplished by extending the minimal hypersurface approach in the presence of singularities and controlling the singular sets which arise. The work is in a recent paper which is joint with S. T. Yau.

Robert Wald (Chicago)

Title: Canonical Energy

Mu-Tao Wang (Columbia)

Title: Linear stability of Schwarzschild black hole: the Cauchy problem for metric coefficients

Abstract: The Schwarzschild solution of the vacuum Einstein equation in general relativity is the unique static solution that represents an isolated gravitating system of a single black hole. Studies, both theoretically and experimentally, of such a system are modeled on the Schwarzschild solution and its perturbations. The stability of the Schwarzschild solution is thus of utmost importance. I will address the linear stability of the Schwarzschild solution, which has a long history and rich literature involving the works of both physicists and mathematicians, and culminating in the recent breakthrough of Dafermos-Holzegel-Rodnianski. In joint work with Pei-Ken Hung and Jordan Keller, we provide a different and simpler proof that reveals the underlying geometric structure of the vacuum Einstein equation at a more elementary level.

Neshan Wickramasekera (Cambridge)

Title: Regularity of stable CMC hypersurfaces

Abstract: Recent combined work of Guaraco, Hutchinson, Tonegawa and the speaker has lead to an alternative approach to the classical Almgren-Pitts-Schoen-Simon existence theory for minimal hypersurfaces. This new method produces a minimal hypersurface, in any given compact Riemannian manifold, as a weak limit of level sets of solutions to a family of Allen--Cahn equations. The approach decouples the existence and regularity parts in a clean way, placing a heavier burden on the latter. In particular the construction part of the method is PDE theoretic and avoids the intricate geometric min-max argument of Almgren and Pitts, and for regularity conclusions, it relies on a sharp varifold regularity theory that builds on the Schoen-Simon compactness theory for stable minimal hypersurfaces with small singular sets. In this talk, after briefly outlining this new approach as motivation, we will focus on the following: a sharp regularity and compactness theory for stable CMC varifolds (joint work with Costante Bellettini) that extends the minimal hypersurface regularity theory.