The Bounded L2 Curvature Conjecture

Details The Bounded L2 Curvature Conjecture The Bounded L2 Curvature Conjecture

2012-04-08

arxiv.org/abs/1204.1767v1

Mathematics

Analysis of PDEs

Scientific paper

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 another, more subtle, scaling tied to its causal geometry; i.e. $L^2$ 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 \cite{Ba-Ch1}, \cite{Ba-Ch2}, \cite{Ta1}, \cite{Ta2}, \cite{Kl-R1} and optimized in \cite{Kl-R2}, \cite{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. In this first paper 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 \cite{param1}-\cite{param4}. The full proof of our main theorem relies on these results

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