2008
DOI: 10.1007/s00220-008-0412-x
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Injectivity Radius of Lorentzian Manifolds

Abstract: Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer (p, T) -where p is a point of the manifold and T is a future-oriented time-like unit vect… Show more

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Cited by 26 publications
(44 citation statements)
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“…They consider a net of smooth Riemannian metrics g ǫ obtained from g through convolution with a mollifier, and use methods from comparison geometry to obtain sufficiently strong estimates on the exponential maps of the regularized metrics, so as to be able to carry over the bi-Lipschitz property through the limit. In order to perform this last step in the general pseudo-Riemannian case they use some results on comparison geometry for indefinite metrics recently obtained by Chen and LeFloch [14]. They also show that the Riemannian case can be dealt with using the Rauch comparison theorem.…”
Section: Remarkmentioning
confidence: 99%
“…They consider a net of smooth Riemannian metrics g ǫ obtained from g through convolution with a mollifier, and use methods from comparison geometry to obtain sufficiently strong estimates on the exponential maps of the regularized metrics, so as to be able to carry over the bi-Lipschitz property through the limit. In order to perform this last step in the general pseudo-Riemannian case they use some results on comparison geometry for indefinite metrics recently obtained by Chen and LeFloch [14]. They also show that the Riemannian case can be dealt with using the Rauch comparison theorem.…”
Section: Remarkmentioning
confidence: 99%
“…There are no natural useful notions of convexity radius or injectivity radius of a Lorentzian manifold (M , g ), but one can define such radii via an auxiliary Riemannian metric η on M : the "size" of subsets of the domain of exp g x in each tangent space T x M can be measured in terms of η. The resulting "mixed" injectivity radius has been studied by Chen-LeFloch [7] and Grant-LeFloch [12] in the situation when η has the form Wick(g , t ) for some temporal function t , as in our Theorem 1.9. Example 3.9 suggests that statement (ii) of Theorem 1.8 becomes true for an arbitrary semi-Riemannian metric g 0 and an arbitrary additional Riemannian metric η if one drops the completeness claim and replaces the (undefined) radius conv g 0 [u] (resp.…”
Section: Theorem Let F Be a Foliation On A Manifold M Let G Be A Rmentioning
confidence: 99%
“…This yields sec g [u] L (σ) = Riem g [u] L (e 1 , e 2 , e 1 , e 2 ) ≤ Riem g [u] L g [u] L . Since z lies in K i +1 \ K i −2 , the definition of Φ 0 (u) implies (7). It remains to check (8).…”
Section: Theorem Let F Be a Foliation On A Manifold M Let G Be A Rmentioning
confidence: 99%
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“…In other words, one can associate a stationary Riemannian metricĝ to a stationary Lorentzian metric g M with the same Killing field. See [4] and [5] for similar ideas in treating the injectivity radius estimate and local optimal regularity of Einstein spacetimes. Our local curvature or gradient estimates are the followings:…”
Section: Introductionmentioning
confidence: 99%