부산대학교 도서관

The problem of estimating injectivity radii as well as comparing curvature and volume are classical subjects in Riemannian geometry, which face up to serious difficulties when they are considered in the Lorentzian background. These problems come from the indefinite metric and its algebraic consequences on the different kinds of causal character 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 Lorentzian intrinsic conditions, some of them being global, which allow us to get comparison 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)]. \par When the Lorentz manifold $(M,g)$ is seen as a result of the Einstein evolution equations, it seems natural to fix a timelike vector field $T$, consider the Riemannian metric $g_T$ naturally associated 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 to the $g_T$-ball ${\scr B}_T(p,r)$. \par Even though Inj$_g(M,p,T)$ estimates are local and non-intrinsic, they allow us to build coordinate charts of enough regularity, which become important for studying the regularity of the Lorentzian metric [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 this fact, the paper under review states different injectivity radius estimates at either a point or its past lightcone. \par In particular, the following is shown. Let $(M^{n+1},g)$ be a time-orientable Lorentzian $(n+1)$-manifold. Let $(p,T)$ be an observer consisting of a point $p \in M$ and a future-oriented timelike unit vector $T \in T_pM$. Assume that the exponential map exp$_p$ is defined in a ball $B_T(0,r) \subset T_pM$ and the Riemann curvature 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 a vector in the Riemannian ball $B_T(0,r) \subset T_pM$. Then there exists a constant $c(n)$, depending only on the dimension of the manifold, such that $$ \frac{{\rm Inj}_g(M,p,T)}{r} \geq c(n) \frac{{\rm Vol}_g(\exp_p(B_T(p,r))(p,c(n)r))}{r^{n+1}}. $$ \par Regularity in harmonic coordinates as well as extensions to indefinite metrics with general signature are also detailed.