International audienc

This is the main paper in a sequence in which we give a complete proof of the bounded L 2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L 2 -norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, L 2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1 + 1 (based on Strichartz estimates) were obtained in [2], [3], [49], [50], [19] and optimized in [20], [37], the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role.To achieve our goals we recast the Einstein vacuum equations as a quasilinear so(3, 1)valued Yang-Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including L 2 error bounds which is carried out in [42]-[45], as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in [46]. It is at this level that our problem is critical. Indeed, any known notion of a parametrix relies in an essential way on the eikonal equation, and our space-time possesses, barely, the minimal regularity needed to make sense of its solutions.1 The original proof in [5] relied on representation formulas, following an approach pioneered by Sobolev, see [38].2 Based only on energy estimates and classical Sobolev inequalities. 3 The original proof in [12], [14] actually requires one more derivative for the uniqueness. The fact that uniqueness holds at the same level of regularity than the existence has been obtained in [33] 4 That is any past directed, in-extendable causal curve in M intersects Σ 0 .11 Maximal foliation together with spatial harmonic coordinates on the leaves of the foliation would be the coordinate condition closest in spirit to the Coulomb gauge. 12 We would like to thank L. Andersson for pointing out to us the possibility of using such a formalism as a potential bridge to [16] .13 Note also that additional error terms are generated by projecting the equations on the components of the frame.

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.