CONTRAINTES D’EINSTEIN:

passé, présent et futur

EINSTEIN CONSTRAINTS: past, present, and future

Erwann DELAY (Avignon), Philippe G. LeFLOCH (Sorbonne), Romain GICQUAUD (Tours)

Research project ANR-23-CE40-0010-02 (Einstein-PPF)

funded by the Agence Nationale de la Recherche (ANR)

MEMBRES / MEMBERS

Xavier Bekaert (Tours), Erwann Delay (Avignon), Alberto Farina (Amiens), Yuxin Ge (Toulouse), Romain Gicquaud (Tours), Eric Gourgoulhon (Meudon), Emmanuel Hebey (Cergy-Pontoise), Marc Henneaux (Collège de France), Marc Herzlich (Montpelier), Emmanuel Humbert (Tours), Paul Laurain (Sorbonne, IMJ), Bruno Le Floch (Sorbonne, LPTHE), Philippe LeFloch (Sorbonne, JLL), Simon Raulot (Rouen), Emmanuel Trélat (Sorbonne, JLL), Caterina Vâlcu (Créteil)

POSTDOCS and Ph.D. STUDENTS

— Fellowship in Avignon : now closed

— Fellowship in Tours : currently open Apply HERE

— Fellowship in Paris : to be open in 2026. Email: contact [AT] philippelefloch [DOT] org

ACTIVITÉS RECENTES / RECENT ACTIVITIES

— 27 au 29 mai 2024: Rencontre à Avignon, organisée par Erwann Delay

— 11 au 13 juin 2025 : Conférence à Paris organisée par Philippe LeFloch, dans le cadre de la Conférence Lichnerowicz 2025

— Rencontres en ligne organisées tous les semestres

Alberto FARINA (Rouen)

Résultats de classification, théorèmes de rigidité et EDPs semi-linéaires sur les variétés riemanniennes : une approche par p-fonction

Résumé. Nous considérons les solutions d’équations elliptiques semi-linéaires critiques et sous-critiques sur des variétés riemanniennes complètes, non compactes et étudions leur classification ainsi que l’effet de leur présence sur la variété sous-jacente. Lorsque la courbure de Ricci est non-négative, nous prouvons à la fois la classification des solutions positives à l’équation critique et la rigidité de la variété ambiante. Les mêmes résultats sont établis pour les solutions de l’équation de Liouville sur les surfaces riemanniennes. Nos résultats sont obtenus en montrant qu’une fonction auxiliaire appropriée (P-fonction) est constante. Ceci implique la classification à la fois des solutions et de la variété sous-jacente. L’analyse effectuée sur la fonction P permet également de classifier les solutions non-négatives d’équations sous-critiques sur les variétés qui vérifient une inégalité de Sobolev et une condition d’intégrabilité sur la partie négative de la courbure de Ricci.

Romain GICQUAUD (Tours)

Sur le “mass aspect” des variétés asymptotiquement hyperboliques

Abstract. Il existe deux définitions de la masse pour les variétés asymptotiquement hyperboliques. La première due à P. Chrusciel et M. Herzlich est un analogue de la définition ADM classique, i.e. une intégrale sur une sphère de rayon infini d’une certaine quantité construite à partir de la métrique et de ses dérivées premières. Mais il existe une autre définition due à M.-T. Wang qui décrit la masse comme l’intégrale du premier terme non nul dans l’expansion asymptotique de la métrique (communément appelé mass aspect). Je montrerai que ce mass aspect admet une définition dans un cadre de régularité faible analogue à celui de Chrusciel-Herzlich et qu’il possède de bonne propriétés de covariance sous les changements de carte à l’infini. Ce travail est en collaboration avec Anna Sakovich (Université d’Uppsala).

Klaus KRÖNCKE (Stockholm)

Dynamical stability and instability of Poincaré Einstein manifolds

Abstract. We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and I established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics. This is joint work with Louis Yudowitz.

Benjamin MECO (Uppsala)

The generalized Jang equation in the asymptotically anti-de Sitter setting and possible applications

Abstract. 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. Furthermore, we will show how to construct a geometric solution of the generalized Jang equation in this setting, in the case when the dimension is less than 8 and for very general asymptotics, using methods from geometric measure theory.

Pieralberto SICBALDI (Granada)

A Schiffer-type problem in annuli and applications to Euler flows

Abstract.  If on a smooth bounded domain of the plane there is a (non-constant) Neumann Laplace eigenfunction that is locally constant on the boundary, must the domain be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture. In fact, here the eigenfunction is allowed to take a different constant value on each connected component of the boundary. Many of the known rigidity properties of the original Schiffer problem are essentially preserved. In this talk I will show that the answer to such question is negative by constructing
a family of nontrivial doubly connected domains with the above property. Furthermore, I will show that this implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial. This talk is based on a joint work with A. Enciso, A. J. Fernández and D. Ruiz.

Caterina VÂLCU (Villetaneuse)

Solving initial data for Kaluza-Klein spacetimes

Abstract. We study the constraint equations for Einstein equations on manifolds of the form Rn+1×Tm, where Tm is a flat m-dimensional torus. Spacetimes with compact directions were introduced almost a century ago by Theodor Kaluza and Oskar Klein as an early attempt of unifying electromagnetism and general relativity in a simple, elegant way. The aim of this article is to construct initial data for the Einstein equations on manifolds of the form Rn+1× Tm, which are asymptotically flat at infinity, without assuming any symmetry condition in the compact direction. We use the conformal method to reduce the constraint equations to a system of elliptic equation and work in the near CMC (constant mean curvature) regime. The main new feature of the proof is the introduction of new weighted Sobolev spaces, adapted to the inversion of the Laplacian on product manifolds. Classical linear elliptic results need to rigorously proved in this new setting. This is joint work with Cécile Huneau.

MAIN OBJECTIVES OF THIS PROJECT

Parametrization of initial data sets for the Einstein equations. The project focuses on the global geometry of Riemannian manifolds satisfying Einstein constraints arising in general relativity. In other words, we are interested in the geometric and analytic properties of initial data sets, consisting of a spacelike hypersurface (representing a slice of “present” time) in a spacetime satisfying Einstein’s field equations (possibly coupled to matter fields). Among our main objectives, we will seek a parametrization of “all” such hypersurfaces and describe their global geometric and asymptotic properties, for instance their behavior at spacelike infinity or in the vicinity of gravitational singularities. We intend to encompass a broad variety of geometric setups, including compact as well as non-compact manifolds with, for instance, asymptotically Euclidian or asymptotically hyperbolic ends. Interestingly, many central concepts of geometry and general relativity play a role.

Seek a unification of techniques and results. Our project takes its roots in our past, namely Lichnerowicz’s pioneering work on the so-called Conformal Method, later generalized by many researchers (see below). The Project also builds upon present developments on the so-called Variational Method, first proposed by Corvino and Schoen. We also intend to contribute to shape the future in this field and seek a unification of the results and methods currently available on Einstein’s constraint equations. Despite these old and new advances on the subject, including significant contributions by members of this Project, the literature on Einstein constraints remains a collection of dispersed results and ad-hoc techniques, and still offers many outstanding open problems: definition of asymptotic invariants, rigidity properties, asymptotic behaviors, etc.

Existing and new directions. To the opinion of the members of this Project, in past years this field has not received sufficient attention (nor funding), and given all recent developments (on the evolution problem for the Einstein equations, on the study of curvature invariants in Riemann geometry, and in numerical relativity) the time is now ripe to fill this gap and push the research in geometric analysis in the proposed directions. This research Project should also open up new directions on modified theories of gravity (such as the f(R,T) theory and Kaluza-Klein theory) and numerical relativity.

RECENT CONTRIBUTIONS by the members of the project

 (To be completed)