Schedule for: 18w5108 - Asymptotically Hyperbolic Manifolds

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will begin by reviewing the renormalized volume for Poincaré-Einstein metrics. I will then discuss an analogous construction of renormalized volume for singular Yamabe (a.k.a. Loewner-Nirenberg) metrics. I will conclude by describing joint work with Matt Gursky concerning an unexpected invariance property of the renormalized volume for singular Yamabe metrics in dimension $4$ related to the Chern-Gauss-Bonnet formula and related phenomena for solutions of a singular $\sigma_2$ Yamabe problem.

Let $M$ be a compact oriented $d$-dimensional manifold with boundary $N$. A natural geometric boundary value problem is to find an asymptotically hyperbolic Einstein metric $g$ on (the interior of) $M$ with prescribed `conformal infinity’ on $N$. A little more precisely, the problem is to find (Einstein) $g$ with the boundary condition $x^2g$ tends to a metric $h$ on $N$ as $x$ goes to $0$, $x$ being a boundary defining function for $N$. The freedom to rescale $x$ by an arbitrary smooth positive function means that only the conformal class of $h$ is naturally well defined. Hence the terminology `conformal infinity’ in this boundary problem. Since the pioneering work of Graham and Lee (1991) the problem has attracted attention from a number of authors. If the dimension $d$ is $4$, there is a refinement, asking that $g$ be anti-self-dual as well as Einstein (satisfying the same boundary condition). If $M$ is the ball, this is the subject of the positive frequency conjecture of LeBrun (1980s) proved by Biquard in 2002. In this talk, which is based on joint work with Joel Fine and Rafe Mazzeo, I shall explain a gauge theoretic approach to the ASDE problem which is readily applicable for general $M$ and the currently available results.

Sasaki geometry is often viewed as an odd dimensional analogue of Kaehler geometry. In particular a Riemannian or pseudo-Riemannian manifold is Sasakian if its standard metric cone is Kaehler or, respectively, pseudo-Kaehler. We show that there is a natural link between Sasaki geometry and projective differential geometry. The situation is particularly elegant for Sasaki-Einstein geometries and in this setting we use projective geometry to provide the resolution of these geometries into “less rigid” components. This is analogous to usual picture of a Kaehler structure: a symplectic manifold equipped also with a compatible complex structure etc. However the treatment of Sasaki geometry this way is locally more interesting and involves the projective Cartan or tractor connection. This enables us to describe a natural notion of compactification for complete non-compact pseudo-Riemannian Sasakian geometries. For such compactifications the boundary at infinity is a conformal manifold with a Fefferman space structure—so it fibres over a CR manifold. This is nicely compatible with the compactification of the Kaehler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. This is joint work with Katharina Neusser and Travis Willse.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Given a class of conformally compact Einstein manifolds with boundary, we are interested to study the compactness of the class under some local and non-local boundary constraints. I will report some joint work with Yuxin Ge and some recent improvements under discussion also with Jie Qing of the problem on the $3+1$ setting.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Holography relates gravity in asymptotically locally hyperbolic manifolds to conformal field theories in one less dimension. This talk will focus on recent developments in holography that may be of particular interest to mathematicians and relativists. Topics covered will include (i) generalisations of holography to different classes of spacetime asymptotics (ii) minimal surfaces in hyperbolic geometries, their renormalised areas and entanglement and (iii) relation between Bondi-Sachs and Fefferman-Graham analysis of asymptotic structure.

We present some results concerning the mass aspect and positivity of mass for asymptotically locally hyperbolic (ALH) manifolds. This is based on joint work Piotr Chrusciel, Luke Nguyen and Tim Paetz.

The mass of an asymptotically hyperbolic manifold is a vector in Minkowski space defined in terms of the geometry at infinity of the manifold. It enjoys covariance properties under the change of coordinate chart at infinity. In this talk we classify covariants satisfying similar properties. This is a join work with J Cortier and M Dahl.

In many situations in Newtonian gravity, understanding the motion of the center of mass of a system is key to understanding the general "trend" of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in general relativity. However, while the definition of the center of mass via the mass density is straightforward in Newtonian gravity, there is a priori no definitive corresponding notion in general relativity, let alone in the asymptotically hyperbolic setting. I will present a geometric approach to defining the center of mass of an asymptotically hyperbolic initial data set, using foliations by constant mean curvature near the asymptotically hyperbolic end of the initial data set. This approach is joint work with Cortier and Sakovich, builds upon work by Neves and Tian, and extends results in the asymptotically Euclidean case going back to Huisken and Yau.

In this talk I will report our recent works on convex hypersurfaces in hyperbolic space. To study hypersurfaces in hyperbolic space analytically, one needs to find ways to parametrize it, preferably globally. We consider two parametrizations: vertical graph and hyperbolic Gauss map. To get a global parametrization, one needs understand the interrelation of convexity and embeddedness. It is also important to understand the asymptotic of the geometry at ends. In this talk I will report some of our recent works on global and asymptotic properties of hypersurfaces with nonnegative sectional curvature or Ricci curvature in hyperbolic space, where our use of $n$-Laplace equations seems to be new.

Given a metric $g$ on the $2$-sphere $S^2$ with Gaussian curvature bound below by $-3$, and non-negative constant $H$, we construct asymptotically hyperbolic manifolds whose boundary is isometric to $(S^2, g)$ and has mean curvature $H$ (with respect to the inward-pointing unit normal). These AH manifolds have mass that is controlled in terms of $g$ and $H$, reducing to the hyperbolic Hawking mass of $(S^2, g, H)$ as $g$ becomes round or $H$ tends to zero. This gives an upper bound for an asymptotically hyperbolic analogue of the Bartnik mass. The construction is based on work of Mantoulidis and Schoen, where they used similar ideas to effectively compute the (usual AF) Bartnik mass of apparent horizons. This is joint work with Armando Cabrera Pacheco and Carla Cederbaum.

The rigidity of the Riemannian positive mass theorem asserts that the ADM mass of an asymptotically flat manifold with non-negative scalar curvature equals zero if and only if the manifold is the Euclidean space. It is natural to ask if the ADM mass of a given manifold is close to zero, is the manifold close to the Euclidean space in some sense? Huang and Lee proved the stability (in the sense of currents) of the positive mass theorem for asymptotically flat graphs. We will describe how to use results of Dahl, Gicquaud and Sakovich to adapt Huang and Lee's ideas to obtain a stability result for positive mass theorem for asymptotically hyperbolic graphs.

In this talk I'll survey results concerning the normalized Ricci flow evolving from a conformally compact asymptotically hyperbolic metric. I'll discuss joint work with Mazzeo and Woolgar on the behavior of the renormalized volume along the flow of both asymptotically Poincaré-Einstein metrics and metrics with an even expansion. I'll then discuss joint work with Woolgar on the long-time existence of the flow for rotationally symmetric asymptotically hyperbolic initial data, and ongoing work with Guenther and Isenberg on the stability of these flows.

In this lecture, based on joint work with Peter Hintz, I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in $3+1$ dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum. I will first discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In the second part of the talk I will discuss analytic aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.

The gravity theory given by the Einstein-Hilbert action has a natural extension in higher dimensions known as Lovelock gravity or Gauss-Bonnet gravity. I will discuss imposing these curvature equations on a conformally compact manifold.

We address a geometric inverse problem: Consider a simply connected Riemannian 3-manifold $(M,g)$ with boundary. Assume that given any closed loop \gamma on the boundary, one knows the area of the area-minimizer bounded by \gamma. Can one reconstruct the metric g from this information? We answer this in the affirmative in a very broad open class of manifolds. We will briefly discuss the relation of this problem with the question of reconstructing a metric from lengths of geodesics, and also with the Calderon problem of reconstructing a metric from the Dirichlet-to-Neumann operator for the corresponding Laplace-Beltrami operator. We also raise the analogous question for asymptotically hyperbolic manifolds, and the significance of their question in physics. Joint with T Balehowsky and A Nachman.

There have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. This talk will begin with introducing several results obtained by this foliation method. I will talk about my work on a particular construction of asymptotically hyperbolic Riemannian $3$-manifolds using the Ricci flow solution on a closed surface as a foliation. This can be used to get an asymptotically hyperbolic extension from a closed surface with some condition. The talk is based on the preprint https://arxiv.org/abs/1802.01019.

I will present new results on the behavior of extremal surfaces in general asymptotically AdS spacetimes. These include new theorems on the maximal extent of boundary-anchored extremal surfaces and a new singularity theorem. Based on work in progress with Daniel Harlow.

In this talk, we discuss whether a conformal class on the boundary $M$ of a given compact manifold $X$ can be the conformal infinity of a Poincaré-Einstein metric in $X$. We construct an invariant of conformal classes on the boundary $M$ of a compact spin manifold $X$ of dimension $4k$ with the help of the Dirac operator. We prove that a conformal class cannot be the conformal infinity of a Poincaré-Einstein metric if this invariant is not zero. Furthermore, we will prove this invariant can attain values of infinitely many integers if one invariant is not zero on the above given spin manifold. This talk is based on a joint work with Gursky and Stolz.

It has been conjectured that anti-de Sitter spacetime is unstable. After briefly presenting some evidence supporting this conjecture, I will discuss various aspects of the problem in a broader context of recent developments in spatially confined nonlinear dispersive equations.

Hyperboloidal initial data sets do not generally evolve into spacetimes with a well-defined Scri unless they satisfy the shear-free condition. In this talk we present a procedure for gluing shear-free hyperboloidal initial data which leaves the conformal infinity connected. Our work is done within the framework of weakly asymptotically hyperbolic metrics, and a special attention is paid to the shear-free condition. This is a collaboration with Paul T Allen, James Isenberg and John M Lee.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.

The Yamabe invariant of an asymptotically Euclidean (AE) manifold is defined analogously to that of a compact manifold. Nevertheless, the prescribed scalar curvature problem in the AE setting has features that are quite different from its compact counterpart. For example, a Yamabe positive AE manifold admits a conformally related metric that has a scalar curvature with any desired sign: positive, negative or zero everywhere. In this talk we discuss the resolution of the prescribed nonpositive scalar curvature problem for AE manifolds and its application to general relativity.