In this talk we intend to present some advances in the study of the Einstein constraint equations (ECE) based on two recent papers [1, 2]. The main subject of these papers is the analysis of elliptic systems arising within the study of the ECE through the conformal method, mainly on non-compact manifolds and in situations where such systems are completely coupled. Contrary to the case of closed manifolds under decoupling conditions, on non-compact manifolds and/or under coupling conditions the understanding of these systems is far from complete, even for some cases extremely relevant for physics, such as the coupling between gravity and a charged fluid. In this context, during this talk, our aim is to present existence results for the conformally formulated ECE which allow both for these types of interactions with matter fields as well as for far from constant mean curvature conformal data. These results can be produced in great generality in the case that the non-compact manifolds have controlled asymptotic geometry, for instance, for asymptotically Euclidean manifolds. Also, several results can be extended through more subtle techniques to general complete manifolds. In particular, an existence criterion based on the existence of a priori barriers holds for the coupled system with electromagnetic sources in the more general situation. Although the existence of such barriers on general complete manifolds is not settled, for conformal data of bounded geometry we have constructed such barriers in [2]. In the fully coupled situation, some regularity issues which can be of interest for the evolution problem associated to solutions of the ECE also appear, mainly when trying to produce *H ^{s}*-initial data. If time allows for it, we intend to briefly comment on these regularity issues as well.

- R. Avalos and J. H. Lira,
*Einstein-Type Elliptic Systems*, Ann. Henri Poincaré**23**, 3221–3264 (2022). - R. Avalos, J. H. Lira and N. Marque,
*Einstein Type Systems on Complete Manifolds*arXiv:2201.08347v1 (2022).

Here, I present few results from my Master Thesis which are actually published in arxiv. I will focus on the equivalence between three variational principles (called *pictures*) used in general relativity: the metric picture (the most popular case, where the metric is the fundamental object), the Palatni picture (here we control both the metric tensor and the connection) and the affine picture (the least known, where only the connection is controlled). Such construction shows quite an interesting fact: the matter could affect the character of the connection or, in other words, the non-metricity of the connection could be interpreted as a matter field. As an example of all this deliberations I will present the model of gravity with the cosmological constant and proposition of unification of the gravity and electromagnetism. Such a model embraces a few classical ideas: Einstein's intuition that gravity is described by the symmetric part and electromagnetism by the skew-symmetric part of a certain geometric object (here: the Ricci tensor), Weyl's concept of non-metric connection, where the measure of *non-metricity* is played by the electromagnetic potential, Born-Infeled generalisation of electromagnetism, where the "standard" Maxwellian theory arises as its weak field approximation.

A spacetime is said to satisfy the Penrose property if every pair of points on past and future null infinity can be connected by a timelike curve. Penrose showed that this property fails in Minkowski spacetime of any dimension but is satisfied in *3+1* dimensional positive mass Schwarzschild. I will consider the Penrose property in more detail and discuss how it is related to the ADM mass and dimensionality of the spacetime. I will then show how some of the ideas arising in the study of this property can be used to prove a version of the positive mass theorem. Finally, I will discuss how the apparent failure of the Penrose property in higher dimensions (regardless of the ADM mass) may be linked to the greater regularity of possible conformal completions of the spacetime at spacelike infinity.

In this joint work with Cederbaum, Leandro and Dos Santos, we generalize to any dimension *n+1* Robinson's divergence formula used to prove the uniqueness of *(3 + 1)*-dimensional static black holes. To this end, we use a tensor first introduced by Cao and Chen for the analysis and classification of Ricci solitons. We thereby prove the uniqueness of black holes and of equipotential photon surfaces in the class of asymptotically flat *(n+1)*-dimensional static vacuum space-times, provided the total scalar curvature of the horizon is properly bounded from above. In the black hole case, our results recover those of Agostiniani and Mazzieri and partially re-establish the results by Gibbons, Ida, and Shiromizu, and Hwang and finally by Raulot in the case of a spin manifold; in the photon surface case, the results by Cederbaum and Galloway can also be proven. Our proof is not based on the positive mass theorem and avoids the spin assumption.

The well-posedness of standard 2-derivative GR minimally coupled to a variety of matter fields is a bedrock of our understanding of gravity. It was first shown by Choquet-Bruhat in 1952 using the harmonic gauge formulation. However, if we wish to include higher derivative terms in our action, as an effective field theory point of view suggests we should, this formulation no longer provides a well-posed initial value problem. I will discuss the case of Einstein-Maxwell theory modified by higher derivative effective field theory corrections. Field redefinitions can be used to bring the leading parity-symmetric 4-derivative corrections to a form which gives second order equations of motion. I will sketch how to use a recently introduced "modified harmonic" gauge condition to obtain a formulation of these theories which admits a well-posed initial value problem when the higher derivative corrections to the equations of motion are small.

In General Relativity, the notion of singularity is usually defined via either geodesic incompleteness or curvature blow-up. In this talk, I will propose a new definition of singularity using the volume measure on spacetime, and motivate it from a quantum gravity perspective. I will also discuss its pros and cons compared to the usual notions of singularity, both in the Big Bang and in the black hole setting. Based on https://arxiv.org/abs/2305.16995.

The analysis of the (prescribed graphical) mean curvature flow (MCF) is an already quite classical topic in geometric analysis. MCF describes the evolution of an hypersurface along the direction of the mean curvature. In this talk I will give an overview of the MCF of space-like hypersurfaces on space-times arising as graphs of functions. Moreover I will focus on the results concerning the long time existence of mean curvature flows on a class of non-compact generalized Robertson-Walker space-times, obtained jointly with Boris Vertman.

On a Bianchi spacetime, which is the Cauchy development of initial data on a *3*-dim. Lie group, Einstein's equation reduces to a system of ODE's, also when coupled to various matter models, such as a perfect fluid or an electromagnetic field. The 3-dim. Lie groups may be classified according to the Bianchi classification of various types. In vacuum, or coupled to perfect fluids, the dynamics for the higher types VIII and IX are known to have complicated, oscillatory behaviour towards the initial singularity, while for the lower types the initial singularity is understood to be convergent, in the sense that appropriately rescaled state variables converge. In this talk I walk through the proof that generically the nature of the initial singularity of Bianchi type VI_{0} spacetimes is vacuum-dominated, anisotropic and silent.

Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole, it is entirely unclear whether one should expect stability of the family as solutions of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Schwarzschild-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain *close* and dissipate sufficiently fast.

Cosmological observations over the decades favour our universe with a positive cosmological constant. Even a tiniest value of positive cosmological constant profoundly alters the asymptotic structure of space-time, forcing a re-look at the theory of gravitational radiation. In particular, we discuss the surprising consequences of gravitational radiation theory in Bondi-Sachs formalism. We also generalize Bondi et. al's celebrated mass-loss formula in presence of positive cosmological constant.

Systems of wave equations may fail to be globally well-posed, even for small initial data. Attempts to classify systems into well-, and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are only good for categorising behaviour close to null infinity and propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. Furthermore, we give an example of a system satisfying the weak null condition with global ill-posedness due to undifferentiated terms.

One of the major open problems in general relativity is the classification of higher dimensional black holes. In this talk I will focus on smooth, supersymmetric (hence extremal), asymptotically flat black holes of five-dimensional minimal supergravity. All previously known examples of this class possess a biaxial *U(1) ^{2}* symmetry (on top of stationarity). In contrast, here I will assume only a single axial symmetry which "commutes" with the remaining supersymmetry. With these assumptions it is possible to provide a classification of such black hole solutions. In particular, one can show that these solutions have Gibbons-Hawking base of multi-centred type, and the associated harmonic functions on

We show how some results in geometric analysis combined with the Choquet-Bruhat-Geroch theorem yield globally hyperbolic spacetime developments that necessarily contain incomplete null geodesics. We demonstrate this in the Gannon-Lee setting and the cosmological setting.

It is well-known that spacetime expansion can suppress shock formation in fluids. This result, first established in the Newtonian cosmological case, has subsequently been extended to the general relativistic framework for various equations of state and expanding spacetimes. Of particular interest is the case of exponentially expanding FLRW Einstein-Euler spacetimes with a linear equation of state *p=Kρ*. The stability of these solutions in the expanding time direction has been proven for the parameter range *0<K <1/3* by Rodniaski and Speck, although influential work by Rendall suggested that beyond this range inhomogeneous features developing in the fluid density would cause the density contrast to blow up, leaving stability for *K>1/3* in doubt. This was further supported in the form of heuristic arguments by Speck. Recently, however, the work of Oliynyk proved the existence of a class of non-isotropic solutions to the relativistic Euler equations on fixed FLRW spacetimes which are stable for *1/3<K≤1/2*. In this talk we present an extension of Oliynyk’s proof to the range *1/3<K<1* as well as numerical evidence supporting Rendall’s conjecture in the case of isotropic solutions.

The generalized Jang equation was introduced by Bray and Khuri in an attempt to prove the Penrose inequality in the setting of asymptotically Euclidean initial data sets for the Einstein equations. Since then it has appeared in a number of arguments allowing to prove geometric inequalities for initial data sets by reducing them to known inequalities for Riemannian manifolds provided that a certain geometrically motivated system of equations can be solved. We will present a novel argument along these lines that could potentially lead to a proof of the positive mass theorem for asymptotically hyperbolic initial data sets modeling constant time slices of asymptotically anti-de Sitter spacetimes. After that we will discuss how to construct a geometric solution of the generalized Jang equation which is one of the equations in the coupled system underlying the reduction argument, in the case when the dimension is less than 8 and for very general asymptotics.

Friedrich’s framework of the cylinder at spatial infinity introduces a regular initial value problem at spatial infinity for the conformal Einstein field equations. In this formulation, spatial infinity is blown-up to a cylinder that connects the end points of past and future null infinity, these end points are known as the critical sets of null infinity. This representation of spatial infinity can be used to link physical quantities at the critical sets to initial data given on a Cauchy hypersurface. In my work, this formulation is used to evaluate the asymptotic charges for spin-1 and spin-2 fields on Minkowski spacetime and the BMS-supertranslations charges for the gravitational field on a vacuum spacetime. In the case of the spin-1 and spin-2 fields, the asymptotic charges, associated with arbitrary functions on 2-spheres, at the critical sets are shown to be well-defined if and only if the freely specifiable data on the initial hypersurface satisfy certain regularity conditions. It is also shown that the charges can be fully expressed in terms of the free initial data and that there is a correspondence between the charges at past and future null infinity. For the BMS charges, the analysis is more technical but is heavily inspired by Friedrich’s work for the calculation of the Newman-Penrose constants. In my talk, I will discuss the initial result for the spin-1 and spin-2 fields on Minkowski spacetime, and I will also discuss some of the techniques and results obtained so far for the BMS charges on vacuum spacetimes.

A spacetime singularity is called *strong* if any object hitting it is crushed to zero volume. It is called at least *locally naked* if it is a past endpoint of some causal curve in the spacetime manifold. We show the existence of a nonzero measured set of initial data that gives rise to such strong, at least locally naked singularity formed as an end state of an unhindered gravitational collapse of a spherically symmetric inhomogeneous perfect fluid. (Phys. Rev. D 101, 044052, 2020).

Inspired by results from metric geometry, we discuss the globalization of timelike curvature bounds in the Lorentzian pre-length space framework [1]. In particular, utilising the formulation of sectional curvature bounds in terms of triangle comparison, we present a Lorentzian analogue of the Alexandrov patchwork construction for spaces with curvature bounded above. We conclude with a summary of ongoing work on the corresponding problem for Lorentzian pre-length spaces with curvature bounded below and, if time permits, ideas for applications. This work was jointly produced alongside Tobias Beran (Universität Wien), and Felix Rott (Universität Wien) and can be found at [arXiv link TBC].

- M. Kunzinger and C. Sämann,
*Lorentzian Length Spaces*, Annals of Global Analysis and Geometry**54**(2018) 399.

Lorentzian length spaces were put forth by Kunzinger and Sämann in 2018 as the suitable synthetic setting for Lorentzian geometry. Since then, a lot of results from the classical theory of spacetimes have been reproved in this framework. In this talk, which is meant to be an introduction to the topic, we first discuss the basics of Lorentzian length spaces and then go on to mention various recent results and points of interest, such as the splitting theorem.

The Hawking energy is one of the most famous local energies in general relativity, by using a Lyapunov-Schmidt reduction procedure we construct unique local foliations of critical surfaces of the Hawking energy on initial data sets. Any quasilocal energy should satisfy the so-called small sphere limit, therefore we also discuss the relation between these surfaces and the small sphere limit. In particular, we discuss some discrepancies on the small sphere limit, so when approaching a point with these foliations and when approaching as in the small sphere limit.

We show a formula for the ADM mass as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large polyhedra. In the special case of coordinate cubes, we will show an integral formula relating the n-dimensional mass with a geometrical quantity that determines the *(n-1)*-dimensional mass. This is joint work with Pengzi Miao.

General Relativity and Quantum Theory are the two main achievements of physics in the 20th century. Even though they have greatly enlarged the physical understanding of our universe, there are still situations which are completely inaccessible to us, most notably the Big Bang and the inside of black holes: These circumstances require a theory of Quantum Gravity — the unification of General Relativity with Quantum Theory. The most natural approach for that would be the application of the astonishingly successful methods of perturbative Quantum Field Theory to the graviton field, defined as the deviation of the metric with respect to a fixed background metric. Unfortunately, this approach seemed impossible due to the non-renormalizable nature of General Relativity. In this talk, I aim to give a pedagogical introduction to this topic, in particular to the Lagrange density, the Feynman graph expansion and the renormalization problem of their associated Feynman integrals. Finally, I will explain how this renormalization problem could be overcome by an infinite tower of gravitational Ward identities, as was established in my dissertation and the articles it is based upon, cf. arXiv:2210.17510 [hep-th].

The theory of Lorentzian length spaces is a synthetic approach to Lorentzian geometry. It is inspired and motivated by metric geometry, which enabled a synthetic and axiomatic description of Riemmanian geometry. In metric geometry, gluing constructions serve as a powerful and general tool to construct new spaces out of old ones. We adapt these techniques to the synthetic Lorentzian setting and establish some compatibility results. In particular, we present a Lorentzian version of the Reshetnyak gluing theorem, giving preservation of upper curvature bounds under gluing, and we investigate which steps of the causal ladder are preserved under gluing. We conclude the talk with some outlook and applications of these techniques.

Joint work with Tobias Beran.

Based on https://arxiv.org/abs/2201.09695 and https://arxiv.org/abs/2209.06894

In this talk I will show a geometric uniqueness result of the characteristic Cauchy problem in General Relativity. In the standard Cauchy problem, it is known that two initial data *(Σ,h,K)* and *(Σ',h',K')* such that *(Σ,h)* and *(Σ',h')* are isometric and the isometry maps *K'* into *K*, give rise to two isometric spacetimes. In order to prove a result of this type in the case of the characteristic problem one needs an abstract notion of the initial data completely detached from the spacetime one wishes to construct. Such abstract formulation is very recent [1] and relies on hypersurface data, and in particular on double null data. In the first part of this talk the definition of double null data will be reviewed and I will prove that the definition is complete by showing that any double null data can be embedded in some spacetime. If in addition the constraint equations are fulfilled, the double null data turns out to be embeddable in a spacetime solution of the Einstein equations. Then, the necessary conditions for two double null data to give rise to two isometric spacetimes will be determined. This conditions lead to the definition of double null data. I will finish the talk with the proof that two isometric double null data define isometric spacetimes [2]. This gives a geometric uniqueness notion of the characteristic initial value problem in a fully abstract way.

- Mars, M. and Sánchez-Pérez, G.
*Double Null Data and the Characteristic Problem in General Relativity.* - Mars, M. and Sánchez-Pérez, G.
*Covariant definition of Double Null Data and geometric uniqueness of the characteristic initial value problem.*

Dynamical black holes in the non-perturbative regime are not mathematically well understood. Studying approximate symmetries of spacetimes describing dynamical black holes gives an insight into their structure. Utilising the property that approximate symmetries coincide with actual symmetries when they are present allows one to construct geometric invariants characterising the symmetry. In this talk, the generalisation of the construction of a geometric invariant originally due to Dain will be presented. This invariant vanishes on a marginally outer trapped surface if and only if the Killing initial data equations are locally satisfied. Integral to the construction of this invariant is the stability of marginally outer trapped surfaces which will also be discussed.

We define a timelike *4*-vector associated to a spacelike cross section of the standard Minkowski lightcone in *3+1*-dimensions that transforms covariantly under Lorentz boosts of the ambient spacetime, and discuss its connection with a notion of center of mass in the conformal class of the standard *2*-sphere. In particular, any surface of constant curvature embedded into the lightcone corresponds uniquely to a timelike *4*-vector that fully determines the surface. As an application, we obtain quantitive *W ^{2,2}*-estimates between a cross section of the standard lightcone and a surface of constant curvature corresponding to the associated timelike

The question of boundedness of energy from below in general relativity for negative cosmological constant is wide open. For time-symmetric initial data sets, this is the question of whether the energy of asymptotically locally hyperbolic spaces is bounded from below. After giving a short review of the currently known bounds, I describe the construction of asymptotically locally hyperbolic spaces with constant negative scalar curvature, arbitrary high genus, and negative total mass. Based on joint work with Piotr Chruściel and Erwann Delay.