Schedule for: 16w5054 - Geometric Analysis and General Relativity

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.

The notion of isoperimetric mass was introduced by Prof. Huisken about ten years ago, and it has deep relation with ADM mass of an asymptotically flat manifold. In the first part of talk I will discuss non negativity of isoperimetric masses of isoperimetric regions in a 3-dim asymptotically flat manifold with nonnegative scalar curvature. More precisely, I will show that for any isoperimetric region , its isoperimetric mass is always non less than a nonnegative quantity which is in terms of Hawking mass, and it is equal to zero if and only if the AF manifold is isometric to the 3-dim Euclidean space. In the second part of this talk, I will mention an application of isoperimetric mass, i.e. by estimate of isoperimetric masses, I will show that any sequence of isoperimetric regions with enclosed volumes tending to infinity cannot drift off to the infinity on any 3-dim asymptotically flat manifold with nonnegative scalar curvature. If time allowable, I will mention some similar results in the setting of asymptotically hyperbolic case.

We prove that the mass of asymptotically flat 3-manifolds with nonnegative scalar curvature is lower semicontinuous under pointed \(C^0\) Gromov-Hausdorff convergence. In order to prove this, we use Huisken's isoperimetric mass concept, together with a modified weak mean curvature flow argument.

We prove that the mass of asymptotically flat 3-manifolds with nonnegative scalar curvature is lower semicontinuous under pointed \(C^0\) Gromov-Hausdorff convergence. In order to prove this, we use Huisken's isoperimetric mass concept, together with a modified weak mean curvature flow argument.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

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!

For the study of asymptotically flat manifolds in mathematical general relativity, surfaces of constant mean curvature (CMC) have proven to be a useful tool. In 1996, Huisken- Yau showed that any asymptotically flat Riemannian manifold is uniquely foliated by closed CMC surfaces. Since then, several authors have generalized their results in several directions. In this talk, I will discuss how these surfaces characterize the full asymptotic behavior of the surrounding initial data set: A Riemannian manifold is asymptotically flat if and only if it possesses suitable CMC-foliation

There are various ways to assign a Center of mass to an asymptotically Euclidean initial data set with non-zero mass. One relies on a Hamiltonian approach while another way relies on properties of foliations of the asymptotic region by surfaces having constant mean curvature. In this talk, I will first review existing results based on such foliations and the associated Center of mass. I will then describe a recent joint work with C. Cederbaum and A. Sakovich where we extend these results to another foliation giving convergence of the Center of mass for more general situations.

We’ll describe a new theory which is 2/3 of the way from special relativity to general relativity in the sense that it requires 4 functions of t,x,y,z to describe the spacetime metric instead of 6. Flatly foliated relativity is identical to general relativity in spherical symmetry (outside black holes, assuming positive energy density) and still includes gravity, black holes, and the big bang. While the action is the same as general relativity, complete with matter fields, flatly foliated relativity requires spacetime to be foliated by flat, 3 dimensional Euclidean spaces. This extra rigidity prevents gravitational waves, simplifying the theory in some important ways. In particular, the Einstein equation is replaced by an elliptic system of equations on each flat slice, making this theory an interesting stepping stone for understanding general relativity better.

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.

In this talk I will analyze the limit at infinity of the Hawking energy along foliations by spacelike cross sections of an asymptotically flat null hyper surface \(\Omega\). A functional on two-surfaces will be introduced and its monotonicity properties and asymptotic behaviour at infinity will be considered. In particular a foliation in \(\Omega\) will be shown to exist for which a Penrose-like inequality in \(\Omega\) holds where, instead of in terms of the Bondi energy, the area of the MOTS is bounded in terms the asymptotic value of the Hawking energy along the foliation. Applications of the methods for the Penrose inequality will also be mentioned.

We discuss the supremum of the total boundary mean curvature of compact, mean- convex 3-manifolds with nonnegative scalar curvature, with prescribed intrinsic boundary metric. We establish an additivity property for the supremum and exhibit rigidity for maximizers as- suming the supremum is attained. When the boundary consists of topological 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi-Tam and Wang-Yau on the quasi-local mass problem in general relativity. In turn, we define a varia- tional analog of the Brown-York mass without assuming the boundary surface has positive Gauss curvature. This is a joint work with Christos Mantoulidis.

In this talk, we discuss the trapped surface conjecture in terms of the Brown-York mass in conformally flat spaces with equipotential foliations. This is a joint work with E. Malec.

Minimal surfaces are critical points of the area functional. The min-max theory is a variational theory for constructing saddle point type, unstable minimal surfaces. In this talk, we will survey several recent progresses along this direction. We will start by a brief introduction to the basic ideas of this theory. Then we will discuss recent works in some non-compact manifolds and compact manifolds with possibly nonempty boundary.

I will present a general method (pioneered, in special cases, by Ros and Savo) to obtain universal and effective index estimates for minimal hypersurfaces inside a Riemannian manifold, given an isometric embedding of the latter in some (possibly high-dimensional) Euclidean space. This approach can be applied, on the one hand, to tackle a conjecture by Schoen and Marques- Neves asserting that the Morse index of a closed minimal hypersurface in a manifold of positive Ricci curvature is bounded from below by a linear function of its first Betti number, which we settle for a large class of ambient spaces. On the other hand, these methods turn out to be very powerful in studying free-boundary minimal hypersurfaces in Euclidean domains: among other things, we prove a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Applications to compactness theorems, to the explicit analysis of known examples (due to Fraser-Schoen and to Folha-Pecard-Zolotareva) and to novel classification theorems will also be mentioned. This is joint work with Lucas Ambrozio and Benjamin Sharp.

The global study of the space of solutions of the Einstein constraint equations goes back to work of Jerry Marsden and co-workers in the early 1970’s. Since that time the subject has evolved in interesting ways. First it has been possible to localize the deformation theory in certain cases to deform solutions inside a chosen region without changing them outside. A second issue which arises in the deformation theory is a derivative loss problem which occurs when one attempts to place a manifold structure on the constraint manifold of solutions with a finite degree of differentiability. In this general lecture we will give an overview of these issues and developments.

Wine and Cheese reception following the Marsden Memorial Lecture.

We discuss the extension of localized deformation results for the Einstein constraints map to the setting of the dominant energy condition. In the time-symmetric case, the analysis involves the scalar curvature map and the static vacuum equations. The case of the full constraints (joint work with Lan-Hsuan Huang) is more delicate where the dominant energy condition does not hold strictly, and a modified constraint operator is introduced in order to control the first-order change in the dominant energy inequality under perturbation. As time permits, we discuss applications to gluing and density results, and to the Bartnik mass.

We introduce a class of ``weakly asymptotically hyperbolic'' geometries whose sectional curvatures tend to \(-1\) and are \(C^0\) but not necessarily \(C^2\) conformally compact. We establish Fredholm results for geometric elliptic operators in this setting. We use these results to construct constant-mean-curvature solutions to the Einstein constraint equations in the weakly asymptotically hyperbolic setting. We furthermore can ensure that our solutions satisfy the shear-free condition, which is necessary for any spacetime development to admit a regular conformal boundary at future null infinity. This is joint work with James Isenberg, John M. Lee, and Iva Stavrov Allen.

Because of the well-posedness of the Field Equations, one can parameterize the set of all spacetimes by parameterizing the set of all ”initial data” solving the Einstein constraint equations. The standard technique used in this attempt is the conformal method. This method performs admirably for initial data with constant and near-constant mean curvatures. However, more recent results have shown problems for far-from-constant mean curvature data. Notably, last year, The-Cang Nguyen found families of data on Yamabe positive manifolds which exhibit both non-existence and non-uniqueness in that regime, though this result was for a fairly special class of data. Recent (mostly numerical) work with Michael Holst and David Maxwell have shown that these results are much more general, though not universal. In this talk we will discuss these results, their implications, and possible future directions.

Dahl-Gicquaud-Humbert showed that if the mean curvature \(\tau\) has constant sign, at least one of the conformal equations and a certain limit equation has a solution. Based on the idea of this result, we have recently proven that under some certain conditions of \((g,\tau,\sigma)\), there exists a sequence \(\{t_{n}\}\) converging to \(0\) such that the conformal equations associated to \((g,t_{n}\tau,\sigma)\) has at least two solutions. In this short talk, we would like to introduce the half-continuity method applied to the vacuum Einstein conformal constraint equations. More precisely, by using this method, we show that the result of Dahl-Gicquaud-Humbert is still valid for the vanishing \(\tau\), and nonuniqueness results above can be extended to all \(t\) small enough.

I will discuss some recent work concerning the proof of positive mass conjecture for asymptotically hyperbolic manifolds, and, more generally, asymptotically hyperboloidal initial data sets.

In the last 15 years, there has been much progress on higher dimensional solutions to the Einstein equation, much of it from the physics community. They are particularly interesting as, unlike 4 dimensional spacetimes, the horizon is no longer restricted to being diffeomorphic to the sphere, as demonstrated by the celebrated black ring solution of Emparan and Reall. Using the Weyl-Papapetrou coordinates and harmonic map, we show the existence of stationary solutions to the 5 dimensional vacuum Einstein equation, which are bi-axisymmetric solutions with lens space horizons. This is a joint project with Marcus Khuri and Gilbert Weinstein.

It is a well known fact that outermost apparent horizons must allow metrics of positive scalar curvature. It is conceivable that this is also the only restriction on a bounding manifold to be an outermost apparent horizon. In this talk I want to describe some new examples of outermost apparent horizons with non-trivial topology. This is a report on work in progress together with Eric Larsson.

On February 24, 2016 our community suddenly lost a wonderful colleague, collaborator and friend, Sergio Dain. Sergio had been one of the first mathematicians to accept the invitation to participate in this meeting and his absence here this week is deeply felt. We will take a few minute to remember Sergio - with some words from friends and collaborators, and acknowledge his absence.

The classical Penrose inequality gives a variational characterization of Schwarzschild data as that with the minimal mass, amongst all asymptotically flat initial data sets with non- negative scalar curvature and fixed horizon area. A Penrose inequality with charge has also been established, which gives a similar variational characterization of the Reissner-Nordstrom black hole. It has been much more difficult to include angular momentum, and there have been very few results in this direction. Here we present a proof of a Penrose inequality with angu- lar momentum (also including charge), which yields a variational characterization of the Kerr (and Kerr-Newman) data. These techniques are then extended to higher dimensions to obtain Penrose-type inequalities associated with the Myers-Perry black hole (higher dimensional version of Kerr) as well as the Black Ring solution of Emparan and Reall.

In this talk we present in-progress results on geometrical inequalities for axisymmetric rotating objects. We show how the inverse mean curvature flow can be used to obtain an estimate for the size in terms of the angular momentum and the mass of an ordinary object.

In this talk we answer an interesting question about extension of Dain’s inequality to higher dimensions. We prove mass-angular momentum-charge inequalities within the context of vacuum and minimal supergravity. In particular we show the equality holds if and only if the initial data set is isometric to the canonical slice of known stationary solutions.

Einstein’s field equations of General Relativity can be formulated as an initial value problem, where the initial data corresponds to the metric and second fundamental form of a Cauchy hypersurface. This initial value problem has a maximal globally hyperbolic development which is unique up to isometry. The Strong Cosmic Censorship conjecture states that, at least for generic initial data, this development is inextendible, in the sense that there is no solution to the field equations larger than that determined by the initial data. In this talk, I consider the case where the initial data is symmetric under the action of a three- dimensional Lie group (i.e. a Bianchi model), and the stress-energy tensor is assumed to be that of a perfect fluid or vacuum. I present new results proving Strong Cosmic Censorship in a specific class of Bianchi models, namely non-exceptional Bianchi B spacetimes. I further discuss in more detail the asymptotic behaviour of such spacetimes towards the initial singularity.

We consider the initial value problem for the Einstein-massive field system, and for the f(R)-theory of modified gravity, for which the field equations are an extension to the classical Einstein equations. Our main result concerns the nonlinear global stability of Minkowski space- time for both theories. Our proof relies on the Hyperboloidal Foliation Method introduced by the authors to tackle coupled systems of wave-Klein-Gordon equations posed on a curved space (with unknown metric). We are able also to prove that the Cauchy developments of modified gravity converge to those associated with the Einstein equations when the function f(R) approaches R. This is a joint work with Y. Ma (Xi’an) and preprints are available at http://philippelefloch.org/ and ArXiv:1411.4910 and ArXiv:1507.01143.

The Positive Mass Theorem includes a rigidity statement that an asymptotically flat manifold with nonnegative scalar curvature and 0 ADM mass is Euclidean space. The correspond- ing almost rigidity statement that a sequence of asymptotically flat manifolds with nonnegative scalar curvature whose ADM mass converges to 0 should converge in some sense to Euclidean space has only been proven in some settings. I will survey recent results in special settings jointly with Lee, with Huang and Lee, and with Stavrov in which it has been shown that one obtains convergence in the intrinsic flat sense. Intrinsic Flat convergence will be defined and a theorem with Lakzian which allows one to prove intrinsic flat convergence will be presented as well.

In the Newtonian setting the gravitational potential describing a continuous matter distribution can be obtained by integrating gravitational potentials describing point particles. In the relativistic setting point particles can be described by means of Schwarzschild initial data. It is thus plausible that some kind of a limit of a sequence of point particle configurations (such as the configurations considered by Brill and Lindquist) can be taken in order to obtain initial data for relativistic matter models. My talk will demonstrate that such a strategy works in the case of relativistic time-symmetric dust initial data. The crucial step consists of employing the intrinsic flat limit developed by C. Sormani and S. Wenger. The talk is based on a collaboration with C. Sormani.

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a \(C^2\) regular metric. In this talk I will describe how one proves the stronger state- ment that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture.

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.