Convex neighborhoods for Lipschitz connections and sprays

Abstract

We establish that over a C^{2,1} manifold the exponential map of any Lipschitz connection or spray determines a local Lipeomophism and that, furthermore, reversible convex normal neighborhoods do exist. To that end we use the method of Picard-Lindelof approximation to prove the strong differentiability of the exponential map at the origin and hence a version of Gauss' Lemma which does not require the differentiability of the exponential map. Contrary to naive differential degree counting, the distance functions are shown to gain one degree and hence to be C^{1,1}. As an application to mathematical relativity, it is argued that the mentioned differentiability conditions can be considered the optimal ones to preserve most results of causality theory. This theory is also shown to be generalizable to the Finsler spacetime case. In particular, we prove that the local Lorentzian(-Finsler) length maximization property of causal geodesics in the class of absolutely continuous causal curves holds already for C^{1,1} spacetime metrics. Finally, we study the local existence of convex functions and show that arbitrarily small globally hyperbolic convex normal neighborhoods do exist.Comment: 56 pages. v2: modified comment on Gauss lemma under C^2 differentiability conditions on Sect. 1.3. Matches published versio

Hawking’s singularity theorem for C 1,1 -metrics

“…(M, g) is not future timelike geodesically complete. [7,16,17,22], see theorem A.1 to lemma A.8 below, combined with the standard proofs in the smooth case, it is a routine matter to prove the remaining results. So instead of providing full proofs we accurately collect all facts and previous statements entering the respective proofs.…”

“…However, in our proof we will make extensive use of the recent results of C 1,1 -causality theory. An important feature of this paper is that we carefully collect all the results from -C 1,1 causality theory that are required for the proof of the above theorem and show how they can be obtained from [7,17,22]. In addition, in section 4 we make crucial use of causal regularisation techniques to show the existence of maximising curves.…”

“…Recently, however, it has been shown that exp is a bi-Lipschitz homeomorphism. Using a careful analysis of the corresponding ODE problem based on Picard-Lindelöf approximations, as well as an inverse function theorem for Lipschitz maps, Minguzzi [22] established the fact that exp is a bi-Lipschitz homeomorphism [22, theorem 1.11] and used this to derive many standard results in causality theory. Also the present authors in [16, theorem 2.1] and [17] established similar results by extending the refined regularisation methods of [7] and combining them with methods from comparison geometry [5].…”

“…Note that, for j large, (22) implies the right-hand side of (21) to be strictly positive at t = 0. Thus θ − j 1 is initially strictly increasing and θ…”

The Hawking–Penrose Singularity Theorem for C 1,1-Lorentzian Metrics

“…Further, even some standard results from causality theory employed in the smooth proofs have only recently been investigated for metrics of regularity below C 2 . This was most successful in the case where the metric is still C 1,1 , see [32,25,26], but also lower regularities have been treated ( [8,39]). Using these advances and approximation techniques adapted to the causal structure allowed the proof of the Hawking and the Penrose singularity theorem for C 1,1 -metrics (see [27,28]) and, finally, also a proof of the more general Hawking-Penrose Theorem in this regularity could be given, [14].…”