학술논문

Injectivity radius of Lorentzian manifolds.
Document Type
Journal
Author
Chen, Bing-Long (PRC-ZHO) AMS Author Profile; LeFloch, Philippe G. (F-CNRS-LJL) AMS Author Profile
Source
Communications in Mathematical Physics (Comm. Math. Phys.) (20080101), 278, no. 3, 679-713. ISSN: 0010-3616 (print).eISSN: 1432-0916.
Subject
53 Differential geometry -- 53C Global differential geometry
  53C50 Lorentz manifolds, manifolds with indefinite metrics
Language
English
Abstract
The problem of estimating injectivity radii as well as comparingcurvature and volume are classical subjects in Riemannian geometry,which face up to serious difficulties when they are considered in theLorentzian background. These problems come from the indefinite metricand its algebraic consequences on the different kinds of causalcharacter of vectors [see, for instance, L. Å. Andersson and R. E.Howard, Comm. Anal. Geom. {\bf 6} (1998), no.~4, 819--877; MR1664893(2000f:53055)]. However, there are Lorentzianintrinsic conditions, some of them being global, which allow us to getcomparison results [see, for instance, P. E. Ehrlich and M. Sánchez,Tohoku Math. J. (2) {\bf 52} (2000), no.~3, 331--348; MR1772801(2001h:53097); F. Erkekoğlu, E. García-Río and D. N.Kupeli, Gen. Relativity Gravitation {\bf 35} (2003), no.~9, 1597--1615; MR2014496 (2005a:53119)].\parWhen the Lorentz manifold $(M,g)$ is seen as a result of theEinstein evolution equations, it seems natural to fix a timelikevector field $T$, consider the Riemannian metric $g_T$ naturallyassociated to $g$ and $T$, and define the injectivity radius at $p$,${\rm Inj}_{g(M,p,T)}$, as the greatest $r>0$ such that the$g$-exponential map $\exp_p$ is a diffeomorphism relative tothe $g_T$-ball ${\scr B}_T(p,r)$.\parEven though Inj$_g(M,p,T)$ estimates are local and non-intrinsic,they allow us to build coordinate charts of enough regularity, whichbecome important for studying the regularity of the Lorentzianmetric [see M. T. Anderson, J. Math. Phys. {\bf 44} (2003), no.~7,2994--3012; MR1982778 (2004c:53104); S. Klainerman and I.Rodnianski, J. Amer. Math. Soc. {\bf 21} (2008), no.~3, 775--795; 2393426 ]. Motivated by thisfact, the paper under review states different injectivity radiusestimates at either a point or its past lightcone.\parIn particular, the following is shown. Let $(M^{n+1},g)$ be atime-orientable Lorentzian $(n+1)$-manifold. Let $(p,T)$ be anobserver consisting of a point $p \in M$ and a future-orientedtimelike unit vector $T \in T_pM$. Assume that the exponential mapexp$_p$ is defined in a ball $B_T(0,r) \subset T_pM$ and the Riemanncurvature satisfies $\sup_{\gamma}|R_g|_{T_{\gamma}} \leq\frac{1}{r}$, where the supremum is over the domain of definition of$\gamma$ and over every $g$-geodesic $\gamma$ initiating from avector in the Riemannian ball $B_T(0,r) \subset T_pM$. Then thereexists a constant $c(n)$, depending only on the dimension of themanifold, such that$$\frac{{\rm Inj}_g(M,p,T)}{r} \geq c(n)\frac{{\rm Vol}_g(\exp_p(B_T(p,r))(p,c(n)r))}{r^{n+1}}.$$\parRegularity in harmonic coordinates as well as extensions toindefinite metrics with general signature are also detailed.