Overview of the proof of the Bounded $L^2$ Curvature Conjecture

Abstract

This memoir contains an overview of the 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 of 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. 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 \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. The entire proof is obtained in a sequence of 6 papers.

arXiv commentminor modifications. Added chapter 7 on the Strichartz estimates. arXiv admin note: substantial text overlap with arXiv:1204.1767

References in corpus (6)

• Regularity of Wave-Maps in dimension 2+1
• Energy dispersed large data wave maps in 2+1 dimensions
• KIDs are non-generic
• Parametrix for wave equations on a rough background I: regularity of the phase at initial time
• Parametrix for wave equations on a rough background II: construction and control at initial time
• An integral breakdown criterion for Einstein vacuum equations in the case of asymptotically flat spacetimes

Cited by in corpus (7)

• On Lorentzian causality with continuous metrics
• A scattering theory construction of dynamical vacuum black holes
• Gravitational Waves and Their Mathematics
• The Spacetime Geometry of Fixed-Area States in Gravitational Systems
• Hilbert manifold structure for asymptotically hyperbolic relativistic initial data
• On maximal globally hyperbolic vacuum space-times
• Strichartz estimates for gravity water waves