The bounded L2 curvature conjecture

Sergiu Klainerman; Igor Rodnianski; Jeremie Szeftel

doi:10.1007/s00222-014-0567-3

The bounded L² curvature conjecture

Sergiu Klainerman, Igor Rodnianski, Jeremie Szeftel

Mathematics

Princeton University

Research output: Contribution to journal › Article › peer-review

33 Scopus citations

Abstract

This is the main paper in a sequence in which we give a complete proof of the bounded L² curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L²-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, $$L^2$$L2 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 Bahouri and Chemin (Am J Math 121:1337–1777, 1999; IMRN 21:1141–1178, 1999), Tataru (Am J Math 122:349–376, 2000; JAMS 15(2):419–442, 2002), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005), Smith and Tataru (Ann Math 162:291–366, 2005), 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 L² error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768, 2012; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769, 2012; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770, 2012; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771, 2012), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112, 2013). 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.

Original language	American English
Pages (from-to)	91-216
Number of pages	126
Journal	Inventiones Mathematicae
Volume	202
Issue number	1
DOIs	https://doi.org/10.1007/s00222-014-0567-3
State	Published - Oct 1 2015

ASJC Scopus subject areas

General Mathematics

Keywords

35J10

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10.1007/s00222-014-0567-3

Cite this

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abstract = "This is the main paper in a sequence in which we give a complete proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L2-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, $$L^2$$L2 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 Bahouri and Chemin (Am J Math 121:1337–1777, 1999; IMRN 21:1141–1178, 1999), Tataru (Am J Math 122:349–376, 2000; JAMS 15(2):419–442, 2002), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005), Smith and Tataru (Ann Math 162:291–366, 2005), 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 L2 error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768, 2012; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769, 2012; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770, 2012; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771, 2012), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112, 2013). 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.",

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N2 - This is the main paper in a sequence in which we give a complete proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L2-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, $$L^2$$L2 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 Bahouri and Chemin (Am J Math 121:1337–1777, 1999; IMRN 21:1141–1178, 1999), Tataru (Am J Math 122:349–376, 2000; JAMS 15(2):419–442, 2002), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005), Smith and Tataru (Ann Math 162:291–366, 2005), 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 L2 error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768, 2012; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769, 2012; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770, 2012; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771, 2012), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112, 2013). 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.

AB - This is the main paper in a sequence in which we give a complete proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L2-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, $$L^2$$L2 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 Bahouri and Chemin (Am J Math 121:1337–1777, 1999; IMRN 21:1141–1178, 1999), Tataru (Am J Math 122:349–376, 2000; JAMS 15(2):419–442, 2002), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005), Smith and Tataru (Ann Math 162:291–366, 2005), 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 L2 error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768, 2012; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769, 2012; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770, 2012; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771, 2012), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112, 2013). 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.

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