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This article suggests the definition of 'Lorentzian space' weakening the notion of Lorentzian length space just as much that it allows for a functor from the category of causally continuous Lorentzian manifolds to the corresponding category of Lorentzian spaces, and considers three problems in the context of maximal Cauchy developments of Lorentzian length spaces (LLSs): The first is to define pointed Gromov-Hausdorff metrics for spatially and temporally noncompact LLSs, the second to present an explicit non-spacetime example of a maximal vacuum Cauchy development in the LLS category, the third to define canonical representatives for developments. A certain regularity property for geodesics plays a key role in each of the problems., Comment: 10 pages

An interesting recent paper in Nature explores a new method to construct solid bodies rolling along given curves on an inclined plane, based on the Gauss Theorem. The present article complements those examinations with rigorous existence theorems and connections to seemingly unrelated questions like maritime rescue operations., Comment: 8 pages

After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the existence of injective monotonous maps between Lorentzian length spaces. Furthermore we induce metrics on Cauchy subsets, relate respective Hausdorff dimensions, prove existence of rushing Cauchy functions with a given Cauchy zero locus and consider collapse phenomena in this setting., Comment: merged with arXiv:2209.12736 because of close relation and dependency of the contents

We construct analoga of Gromov-Hausdorff space for Lorentzian distances and show a Gromov precompactness result for one of them. After calculating the Dushnik-Miller dimension of Minkowski spaces (of manifold dimension larger than 2) to be countable infinity, we define a dimension for ordered sets recovering the correct manifold dimension, obtain an obstruction for existence of injective monotonous maps between Lorentzian length spaces, induce functorial pseudo-metrics on Cauchy subsets that in the spacetime case coincide with the Riemannian ones, and prove existence of anti-Lipschitz Cauchy functions with a given Cauchy zero locus, a fundamental ingredient for the Sormani-Vega null distance., Comment: 28 pages

We review three examples of functors from Lorentzian categories and their applications in finiteness results, singularity theorems and boundary constructions. The third example is a novel functor from the category of ordered measure spaces to the category of Lorentzian pre-length spaces in the sense of Kunzinger-S\"amann., Comment: 15 pages. The article is a contribution to the conference volume of the conference "Singularity theorems, causality and all that; a tribute to Roger Penrose" (2021) and is in large parts an overview article but also contains some new results

We investigate the Dirac equation in Kerr-Newman space-time, using horizon penetrating coordinates (Eddington-Finkelstein-Coordinates) and the Newman-Penrose formalism to separate the equation into radial and angular systems of ordinary differential equations, and deriving the asymptotics of the radial solutions at infinity and at the Cauchy horizon., Comment: 26 pages

In this article, we show that the map assigning to a metric its restricted holonomy class (conjugacy class of the connected component of the holonomy representation) is lower semicontinuous on the space of $C^2$ metrics on any fixed manifold., Comment: 12 pages

In the context of Bartnik's splitting conjecture, Galloway \cite{G} conjectured that for any globally hyperbolic, spatially compact and future timelike geodesically complete manifold $(M,g)$ satisfying the strong energy condition, the future boundary of $(M,g)$ consists of one point. We show that whereas this statement is true in dimension $2$ due to a result by Ehrlich and Galloway \cite{EG}, it fails in every higher dimension, and that there are plenty of counterexamples, more precisely: On a manifold of dimension $\geq 3$, {\em every} globally hyperbolic conformal class contains future complete metrics satisfying the strong energy condition., Comment: Flawed proof of the main theorem

The article treats some questions around Gromov's filling area conjecture. It intended to show that any filling with volume $< 2 \pi$ would not be attained and to show a local systolic inequality, implying an a priori lower estimate on the total volume. The proof of both assertions, the second of which is wrong, suffered from a computational mistake., Comment: 8 pages. Withdrawn due to a computational error in the derivation of the main results

Via a functor from certain Lorentzian to Riemannian manifolds, we obtain a finiteness result., Comment: 16 pages

On the Geroch-Kronheimer-Penrose future completion IP (X) of a spacetime X, there are two frequently used topologies. We systematically examine $\tau_+ $, the stronger (metrizable) of them, which is the coarsest causally continuous topology, obtaining a variety of novel results, among them a complete characterization of the difference in convergence between both topologies. In our framework, we can allow for X being a chr. space and consequently for the interpretation of IP as an idempotent functor on a category that includes spacetimes of very low regularity. Furthermore, we explicitly calculate $(IP (X), \tau_+ )$ for multiply warped chronological spaces., Comment: 31 pages

We show that any linear quantization map into the space of self-adjoint operators in a Hilbert space violates the von Neumann rule on post-composition with real functions., Comment: 12 pages. The second version changes only the division into chapters and adds a consideration of a related Experiment, the third one restructures the order of the theorems and adds some explanations and references. Theorems and proofs remain unchanged since the first version

Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on $M$ with a parallel spinor gives rise to a solution of the constraint equations on $M\times (a,b)$ (resp. $M\times S^1$)., Comment: Last preprint version. The article is open access, see short link https://rdcu.be/crPr1

We prove Cheeger-Gromov convergence for a subsequence of a given sequence of manifoldswith- boundary of bounded geometry. The method of the proof is to reduce, via height functions, the problem to the setting of Hamilton's compactnes theorem for manifolds without boundary., Comment: 11 pages. Content unchanged since v1, improved presentation

We prove variants of known singularity theorems ensuring the existence of a region of finite lifetime that are particularly well applicable if the solution admits a conformal extension, a property satisfied e.g. by maximal Cauchy developments of Einstein-Maxwell initial values close to the trivial ones., Comment: 22 pages, no figure. Final revised version as submitted to the journal

Given a compact manifold $M$ and a Riemannian manifold $N$ of bounded geometry, we consider the manifold ${\rm Imm} (M,N)$ of immersions from $M$ to $N$ and its subset ${\rm Imm}_\mu (M,N)$ of those immersions with the property that the volume-form of the pull-back metric equals $\mu$. We first show that the non-minimal elements of ${\rm Imm}_\mu (M,N) $ form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics write down the geodesic equation and show local well-posedness in many cases. The question is a natural generalization of the corresponding well-posedness question for the group of volume-preserving diffeomorphisms, which is of great importance in fluid mechanics., Comment: 22 pages, no figure

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have uniformly bounded geometry, we show that both sequences have smoothly Cheeger-Gromov convergent subsequences provided the conformal factors $u_i$ are principal eigenfunctions of an appropriate elliptic operator. Part of our result is a Cheeger-Gromov compactness for manifolds with boundary. We use stable versions of classical elliptic estimates and inequalities found in the recently established 'flatzoomer' method., Comment: 25 pages. The authors discovered a mistake in the paper. In particular, the claim of Theorem B does not hold, however Theorem A still true, and, by insistence of the first author, Theorem A will be published by the second author in a separate paper

We give a simple way to extend index-theoretical statements from partial differential operators with smooth coefficients to operators with coefficients of finite Sobolev order., Comment: 9 pages, no figure

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory., Comment: 20 pages, presentation simplified, result about universal Dirac-operator added

We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. Among other results, we obtain that this property is equivalent to admitting a parallel timelike vector field. We also derive some properties of the space of all such metrics on a given manifold.

The purpose of this article is to present a result on the existence of Cauchy temporal functions invariant by the action of a compact group of conformal transformations in arbitrary globally hyperbolic manifolds. Moreover, the previous results about the existence of Cauchy temporal functions with additional properties on arbitrary globally hyperbolic manifolds are unified in a fully general theorem. The article is written as self-contained as possible, no prior knowledge on Lorentzian geometry is assumed., Comment: 14 pages, no figures

The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples., Comment: 49 pages, LaTeX, 7 figures, minor improvements (published version)

We consider the future causal boundary as a tool to find obstructions to conformal extensions, the latter being a slight generalization to conformal compactifications., Comment: 12 pages, no image

We consider the Cauchy problem of massless Dirac-Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac-Higgs-Yang-Mills theories., Comment: 37 pages, one figure

We show that on every manifold, every conformal class of semi-Riemannian metrics contains a metric $g$ such that each $k$-th-order covariant derivative of the Riemann tensor of $g$ has bounded absolute value $a_k$. This result is new also in the Riemannian case, where one can arrange in addition that $g$ is complete with injectivity and convexity radius greater than 1. One can even make the radii rapidly increasing and the functions $a_k$ rapidly decreasing at infinity. We prove generalizations to foliated manifolds, where curvature, second fundamental form and injectivity radius of the leaves can be controlled similarly. Moreover, we explain a general principle that can be used to obtain analogous results for Riemannian manifolds equipped with arbitrary other additional geometric structures instead of foliations., Comment: 22 pages, 1 figure. The journal article differs from this version only by marginal adaptations required by the publisher's style guidelines, and by one minor typo

We define different notions of black holes, event horizons and Killing horizons for a general time-oriented manifold $(M,g)$ extending previous notions but without the assumption of asymptotical flatness. The notions of 'horizon' are always conformally invariant while the notions of 'black hole' are genuinely geometric. Some connections between the different notions are found. Finally, we put the definitions into the context of the weak cosmic censorship conjecture., Comment: 12 pages

In this short note, a question of patching together globally hyperbolic manifolds is adressed which appeared in the context of the construction of Hadamard states., Comment: 2 pages, submitted to 'Mathematische Zeitschrift'

This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?, Comment: 12 pages, no figure, submitted to Journal of Geometry and Physics. Latest version corrected only the misspelling of the author's name

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings in Minkowski spaces., Comment: 14 pages, no figures

We characterize those spacetimes which admit a isometric (or conformal) embedding in some Lorentz-Minkowski space L^N. In particular, any globally hyperbolic spacetime can be isometrically embedded in L^N. This is proven by a result of its own interest: the construction of a smooth time function whose gradient is bounded away from zero -and, thus, an orthogonal global splitting of the spacetime with bounded lapse. The role of the so-called "folk problems on smoothability" is stressed., Comment: Final version with minor modifications, to appear in Trans. AMS. 13 pages, 1 figure, latex

This note is about a little extension of Nash's embedding theorem in the case of complete manifolds., Comment: last correction due to change of spelling of author's name