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

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

2012-04-08

arxiv.org/abs/1204.1772v1

Mathematics

Analysis of PDEs

Scientific paper

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 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. The entire proof is obtained in a sequence of 6 papers.

Affiliated with

Also associated with

No associations

Rating

Overview of the proof of the Bounded $L^2$ Curvature Conjecture does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Overview of the proof of the Bounded $L^2$ Curvature Conjecture, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Overview of the proof of the Bounded $L^2$ Curvature Conjecture will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-647684