The Bounded L2 Curvature Conjecture
Sergiu Klainerman, Igor Rodnianski, Jeremie Szeftel
TL;DR
This work achieves a complete proof of the Bounded $L^2$ Curvature Conjecture by recasting the Einstein vacuum equations as a covariant $so(3,1)$ Yang–Mills system in a Coulomb-type gauge on a maximal foliation, exposing a null structure that enables bilinear and trilinear spacetime estimates on rough backgrounds. Central to the argument is the construction and control of a parametrix for the wave equation on such backgrounds, together with sharp $L^4$ Strichartz estimates, which hinge on robust control of the null geometry through $L^2$-curvature bounds and volume-radius lower bounds. The strategy reduces to small data via a harmonic-coordinate gluing and rescaling, then closes a bootstrap using propagation of curvature and higher regularity, yielding local existence times depending only on the initial $L^2$ curvature and volume radius. The result clarifies the causal-geometric role of curvature in GR, providing a path toward scale-invariant well-posedness by revealing how $L^2$ curvature governs the injectivity radius and the efficacy of Fourier-integral representations in this quasilinear setting.
Abstract
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 standard 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 [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized in [Kl-R2] [Sm-Ta], 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 \textit{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 [Sz1]-[Sz4], as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in \cite{Sz5}.