A conformal Hopf-Rinow theorem for semi-Riemannian spacetimes

The famous Hopf-Rinow theorem states that a Riemannian manifold is metrically complete if and only if it is geodesically complete. The compact Clifton-Pohl torus fails to be geodesically complete, leading many mathematicians and Wikipedia to conclude that "the theorem does not generalize to Lorentzian manifolds". Recall now that in their 1931 paper Hopf and Rinow characterized metric completeness also by properness. Recently, the author and Garc\'{\i}a-Heveling obtained a Lorentzian completeness-compactness result with a similar flavor. In this manuscript, we extend our theorem to cone structures and to a new class of semi-Riemannian manifolds, dubbed $(n-\nu,\nu)$-spacetimes. Moreover, we demonstrate that our result implies, and hence generalizes, the metric part of the Hopf-Rinow theorem.Comment: 40 page

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality

Global hyperbolicity through the eyes of the null distance

No Hopf-Rinow Theorem is possible in Lorentzian Geometry. Nonetheless, we prove that a spacetime is globally hyperbolic if and only if it is metrically complete with respect to the null distance of a time function. Our approach is based on the observation that null distances behave particularly well for weak temporal functions in terms of regularity and causality. Specifically, the null distances of Cauchy temporal functions and regular cosmological time functions encode causality globally.Comment: Results strengthened and new examples added. Admissible time function renamed to weak temporal function, h-steep renamed to completely uniform. Some minor restructuring of the text. 40 pages, 4 figure

Isomorphisms of algebras of smooth and generalized functions

Ein bekanntes Resultat in der Theorie kommutativer Banachalgebren besagt, dass zwei lokal kompakte Räume $X$ und $Y$ genau dann homöomorph sind, wenn die $C^*$-Algebren der stetigen Abbildungen $C_0(X)$ und $C_0(Y)$ algebraisch isomorph sind. Es ist unser Ziel, analoge Aussagen auch für Algebren glatter Abbildungen bzw. Colombeaualgebren zu zeigen. Die zugrundeliegenden topologischen Räume werden in diesem Fall endlich-dimensionale glatte Mannigfaltigkeiten $X$ und $Y$ sein, die Hausdorff sind und das zweite Abzählbarkeitsaxiom erfüllen. Wir werden sehen, dass nichttriviale multiplikative lineare Funktionale auf $C^{\infty}(X)$ bzw. $\mathcal{G}(X)$ mit Punkten in $X$ bzw. kompakt getragenen verallgemeinerten Punkten $\widetilde{X}_c$ identifiziert werden können. Zudem werden wir beweisen, dass Algebraisomorphismen $C^{\infty}(X) \rightarrow C^{\infty}(Y)$ bereits durch Diffeomorphismen von $Y$ nach $X$ charakterisiert sind. Letzteres gilt sogar für Mannigfaltigkeiten, die das zweite Abzählbarkeitsaxiom nicht erfüllen. Im Zusammenhang mit Colombeau verallgemeinerten Funktionen führt uns diese Fragestellung zu kompakt beschränkten verallgemeinerten Funktionen $\mathcal{G}[Y,X]$, welche die Algebraisomorphismen $\mathcal{G}(X) \rightarrow \mathcal{G}(Y)$ wiederum komplett beschreiben.A well-known result in commutative Banach algebra theory states that two locally compact spaces $X$ and $Y$ are homeomorphic if and only if the $C^*$-algebras of continuous functions $C_0(X)$ and $C_0(Y)$ are algebraically isomorphic. Our aim is to construct a similar theory for algebras of smooth functions and Colombeau generalized functions. The underlying topological spaces are finite-dimensional smooth manifolds $X$ and $Y$ which are Hausdorff and second countable. We find that the non-zero multiplicative linear functions on $C^{\infty}(X)$ and $\mathcal{G}(X)$ can be identified with the points in $X$ and the compactly supported generalized points $\widetilde{X}_c$, respectively. Moreover, we prove that algebra isomorphisms $C^{\infty}(X) \rightarrow C^{\infty}(Y)$ are characterized by diffeomorphisms from $Y$ to $X$, a fact that holds even for manifolds that are not second countable. The same question for Colombeau algebras leads to c-bounded generalized functions $\mathcal{G}[Y,X]$ which again completely determine the algebra isomorphisms $\mathcal{G}(X) \rightarrow \mathcal{G}(Y)$

On the asymptotic behavior of static perfect fluids

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear and polytropic-type equations of state with index n>5. In order to capture the asymptotic behavior we introduce a notion of scaled quasi-asymptotic flatness, which encodes a form of asymptotic conicality. In particular, these spacetimes are asymptotically simple.Comment: 32 pages; minor changes in v2, final versio

Algebras of generalized functions with smooth parameter dependence

We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\widetilde{\mathbb{K}}_{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\widetilde{\mathbb{K}}_{sm}$ and establish some properties of its ideals.Comment: 19 page

Properties of the Null Distance and Spacetime Convergence

The notion of null distance for Lorentzian manifolds recently introduced by Sormani and Vega gives rise to an intrinsic metric and induces the manifold topology under mild assumptions on the time function of the spacetime. We prove that warped products of low regularity and globally hyperbolic spacetimes with complete Cauchy surfaces endowed with the null distance are integral current spaces. This metric and integral current structure sets the stage for investigating spacetime convergence analogous to Riemannian geometry. Our main theorem is a general Lorentzian convergence result for warped product spacetimes relating uniform convergence of warping functions to uniform, Gromov--Hausdorff and Sormani--Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that non-uniform convergence of warping functions in general leads to distinct limiting behavior, such as the convergence of null distances to Gromov--Hausdorff and Sormani--Wenger intrinsic flat limits that disagree.Comment: 58 pages, 9 figures, comments welcome. v2: removed parts of sections 3.3 and 3.4 but the main results and other sections are not affected, minor changes throughou