_________________________________________________________________________________________________________

GRAVITY DAY in PARIS

Recent Advances on Einstein’s Constraint Equations

From 9:00am to 6:00 pm on Sept. 15, 2026 lecture room 15-25-309 (Jussieu)

Laboratoire Jacques-Louis Lions

Sorbonne Université

Organizer: Philippe G. LeFloch Sorbonne Université, Paris

SPEAKERS


Xuantao CHEN (LJJL, Sorbonne)

 Solving the constraint equation for general free data

Abstract. We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo low modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on two-spheres. Our method can be applied to construct various types of initial data, including data with arbitrarily fast decay, borderline decay, and short-pulse type data. This is joint work with Sergiu Klainerman.

Armand COUDRAY (Tours)

Existence and uniqueness of solutions to the conformal constraint equations

Abstract. I will present the results obtained in collaboration with Romain Gicquaud concerning the classical construction of initial data using the conformal method, which was originally proposed by Holst, Nagy and Tsogtgerel and later refined by Maxwell. This method transforms the usual constraint equation for initial data in general relativity into a set of two coupled nonlinear elliptic PDEs. Our work revisits the standard proof by removing certain assumptions. In particular, I will explain how our proof guarantees the uniqueness of solutions to the equations of the conformal method as soon as a bound is imposed on the physical volume and how it provides an explicit construction of solutions.

Romain GICQUAUD (Tours)

On the topology of vacuum initial data in general relativity

Abstract. Parameterizing (vacuum) initial data is a major challenge in general relativity. The most studied method, the conforma method, has been successful in parameterizing the CMC vacuum initial data but beyond it, very limited is known despite decades of efforts. One potential reason is that the parametrizations we are employing are too naive in the sense that the set of seed data has a topology that is too simple compared to the vacuum initial data. Building on earlier works by Jonathan Glöckle and Berndt Amman, we show that there exist compact manifolds of dimension greater than 5 with infinitely many non-trivial homotopy groups. This is a joint work with Jonathan Glöckle.

Bruno LE FLOCH (LPTHE, Sorbonne)

Optimal localization for the Einstein constraints

Abstract. We establish the existence of asymptotically Euclidean initial data sets to Einstein’s vacuum constraints exhibiting both gravitational shielding and arbitrarily low decay. We resolve a conjecture of Carlotto and Schoen on gluing two solutions across an asymptotically conical domain: in the interpolation region we establish optimal estimates at and beyond harmonic decay, together with ADM invariant estimates. Starting from a seed data set that asymptotically solves the constraints, we project it to an exact solution, whose difference from the seed inherits the natural optimal radial decay rate. 

Zhongkai TAO (IHES & Mittag-Leffler, Stockholm)

An h-principle for general relativistic initial data sets

Abstract. I will talk about a flexibility result on the Einstein constraint equations. It uses the Bogovskii-type operators introduced in Mao-Oh-Tao and Isett-Mao-Oh-Tao and generalizes the obstruction-free gluing theorem. This is joint work with Jonathan Luk, Yuchen Mao and Sung-Jin Oh.

Arthur TOUATI (Bordeaux)

Spacelike initial data for black hole stability

Abstract. I will present a joint work with Allen Juntao Fang and Jérémie Szeftel on the resolution of the constraint equations in the context of black hole stability. After having presented the motivations coming from the evolution problem and reviewing the literature on the constraint equations, I will show how a bit of Fredholm theory and the careful construction of a boundary condition for the elliptic system can help avoiding linear obstructions (the famous KIDS) and thus produce initial black hole perturbations with arbitrary decay.

Organized as an event of the ANR project:

Einstein’s constraints: past, present, and future

Romain GICQUAUD (Tours), Erwann DELAY (Avignon), and Philippe G. LeFloch (Sorbonne, Paris)