Abstract
In this paper, we prove a localised version of the bounded \(L^2\)-curvature theorem of (Klainerman et al. Invent Math 202(1):91–216, 2015). More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface \(\Sigma \) with boundary, and show that the time of existence of a classical solution depends only on an \(L^2\)-bound on the Ricci curvature, an \(L^4\)-bound on the second fundamental form of \({\partial }\Sigma \subset \Sigma \), an \(H^1\)-bound on the second fundamental form, and a lower bound on the volume radius at scale 1 of \(\Sigma \). Our localisation is achieved by first proving a localised bounded \(L^2\)-curvature theorem for small data posed on B(0, 1), and then using the scaling of the Einstein equations and a low regularity covering argument on \(\Sigma \) to reduce from large data on \(\Sigma \) to small data on B(0, 1). The proof uses the author’s previous works and the bounded \(L^2\)-curvature theorem as black boxes.
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Notes
On a Lorentzian 4-manifold \(({\mathcal M},\mathbf {g})\), wave coordinates \((x^0,x^1,x^2,x^3)\) satisfy by definition
$$\begin{aligned} \square _{\mathbf {g}} x^{\alpha }=0 \, \text { for } {\alpha }=0,1,2,3. \end{aligned}$$The Einstein equations reduce in wave coordinates to
$$\begin{aligned} \square _{\mathbf {g}} (\mathbf {g}_{{\alpha }{\beta }}) = \mathcal {N}_{{\alpha }{\beta }}(\mathbf {g},{\partial }_\mu \mathbf {g}), \, \text { for } {\alpha },{\beta }=0,1,2,3, \end{aligned}$$where \(\mathcal {N}_{{\alpha }{\beta }}(\mathbf {g}, {\partial }\mathbf {g})\) is a non-linearity that is linear in \(\mathbf {g}\) and quadratic in \({\partial }_\mu \mathbf {g}\), \(\mu =0,1,2,3\).
Note that bounded variation norms are not suitable outside of spherical symmetry. In the absence of spherical symmetry, regularity should be measured with respect to \(L^2\)-based spaces, see [12].
A smooth Riemannian manifold with boundary is called complete if it is complete as a metric space.
Given a cover \((C_i)_{i \in I}\) of \(\Sigma \), its Lebesgue number \(\ell \) is defined as the largest number such that for each point \(p \in \Sigma \), the geodesic ball \(B_g(p,\ell )\) is completely contained in \(C_i\) for some \(i\in I\).
Given a covering \((C_i)_{i=1}^N\) of M, the Lebesgue number \(\ell \) is defined to be the largest real number such that for each point \(p \in M\), there is an \(i\in \{1, \dots , N\}\) such that \(B_g(p,\ell ) \subset C_i\).
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Acknowledgements
This work forms part of my Ph.D. thesis. I am grateful to my Ph.D. advisor Jérémie Szeftel for his kind supervision and careful guidance. This work is financially supported by the RDM-IdF.
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Communicated by P. Chrusciel
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Czimek, S. The Localised Bounded \(L^2\)-Curvature Theorem. Commun. Math. Phys. 372, 71–90 (2019). https://doi.org/10.1007/s00220-019-03458-9
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DOI: https://doi.org/10.1007/s00220-019-03458-9