arXiv subjects

Michael Kunzinger

Publications and source records attributed to Michael Kunzinger.

At least 19 recordsLinked to original sources

arXiv2026

The Hawking Singularity Theorem for H\"older Continuous Metrics with $L^p$-Bounded Curvature

We prove a low-regularity version of Hawking's singularity theorem for Lorentzian metrics in $W^{1,p}$ with Riemann curvature in $L^p$, where $p>2n$ and $n$ the dimension of spacetime. This extends previous results beyond the Lipschitz regime. Under suitable lower Ricci bounds and upper mean curvature assumptions, expressed in terms of temporal functions, we establish both the globally hyperbolic version of Hawking's theorem, in the form of an upper bound on the time separation from a spacelike Cauchy hypersurface, and the version with a compact achronal spacelike hypersurface, yielding timelike RT-geodesic incompleteness. The proof combines regularisations, based on the elliptic RT-equations, to raise the regularity of the metric by one derivative, with a refinement of the previously used manifold convolution. We introduce a new smeared-out notion of mean curvature adapted to the low metric regularity before, and the $W^{2,p}$-hypersurfaces arising after regularisation. As further consequences, we show that $W^{1,p}$-Lorentzian metrics with $L^p$-bounded curvature are causally plain, and we prove a corresponding low-regularity version of Myers's theorem in the Riemannian setting.

gr-qc↗

arXiv2025

Timelike conjugate points in Lorentzian length spaces

We study notions of conjugate points along timelike geodesics in the synthetic setting of Lorentzian (pre-)length spaces, inspired by earlier work for metric spaces by Shankar--Sormani. After preliminary considerations on convergence of timelike and causal geodesics, we introduce and compare one-sided, symmetric, unreachable and ultimate conjugate points along timelike geodesics. We show that all such notions are compatible with the usual one in the smooth (strongly causal) spacetime setting. As applications, we prove a timelike Rauch comparison theorem, as well as a result closely related to the recently established Lorentzian Cartan--Hadamard theorem by Er\"{o}s--Gieger. In the appendix, we give a detailed treatment of the Fr\'{e}chet distance on the space of non-stopping curves up to reparametrization, a technical tool used throughout the paper.

math.DG↗

arXiv2025

Distributional sectional curvature bounds for Riemannian metrics of low regularity

Sectional curvature bounds are of central importance in the study of Riemannian manifolds, both in smooth differential geometry and in the generalized synthetic setting of Alexandrov spaces. Riemannian metrics along with metric spaces of bounded sectional curvature enjoy a variety of, oftentimes rigid, geometric properties. The purpose of this article is to introduce and discuss a new notion of sectional curvature bounds for manifolds equipped with continuous Riemannian metrics of Geroch--Traschen regularity, i.e., $H^1_{\mathrm{loc}} \cap C^0$, based on a distributional version of the classical formula. Our main result states that for $g \in C^1$, this new notion recovers the corresponding bound based on triangle comparison in the sense of Alexandrov. A weaker version of this statement is also proven for locally Lipschitz continuous metrics.

math.DG↗

arXiv2025

Hawking's singularity theorem for Lipschitz Lorentzian metrics

We prove Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type lemma and (2) the replacement of the usual focusing techniques for timelike geodesics -- which in the absence of a classical ODE-theory for the initial value problem are no longer available -- by a worldvolume estimate based on a segment-type inequality that allows one to control the volume of the set of points in a spacelike surface that possess long maximisers.

math.DG↗

arXiv2024

On the completeness of the space $\mathcal{O}_C$

We give a new proof of the completeness of the space $\mathcal{O}_C$ by applying a criterion of compact regularity for the isomorphic sequence space $\lim_{k\rightarrow} (s\hat \otimes (\ell^\infty)_{-k})$. Along the way we show that the strong dual of any quasinormable Fr\'echet space is a compactly regular $\mathcal{LB}$-space. Finally, we prove that $\lim_{k\rightarrow}(E_k\hat \otimes_\iota F) = (\lim_{k\rightarrow} E_k) \hat \otimes_\iota F$ if the inductive limit $\lim_{k \rightarrow}(E_k \hat \otimes_\iota F)$ is compactly regular.

math.FA↗

arXiv2024

Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics

We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity $C^1$. With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the $\mathsf{RCD}$-splitting theorem \cite{gigli2013splitting, gigli2014splitoverview}. Along the way, we establish a Bochner-Weitzenb\"ock identity which permits both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in \cite{mondino2024equivalence}. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see \cite{mondino2024equivalence}) between distributional Ricci curvature bounds and $\mathsf{RCD}$-type bounds, using in part the stability of the variable $\mathsf{CD}$-condition under suitable limits \cite{ketterer2017variableCD}.

math.DG↗

arXiv2024

The equivalence of smooth and synthetic notions of timelike sectional curvature bounds

Timelike sectional curvature bounds play an important role in spacetime geometry, both for the understanding of classical smooth spacetimes and for the study of Lorentzian (pre-)length spaces introduced in \cite{kunzinger2018lorentzian}. In the smooth setting, a bound on the sectional curvature of timelike planes can be formulated via the Riemann curvature tensor. In the synthetic setting, bounds are formulated by comparing various geometric configurations to the corresponding ones in constant curvature spaces. The first link between these notions in the Lorentzian context was established in \cite{harris1982triangle}, which was instrumental in the proof of powerful results in spacetime geometry \cite{beem1985toponogov, beem1985decomposition, galloway2018existence}. For general semi-Riemannian manifolds, the equivalence between sectional curvature bounds and synthetic bounds was established in \cite{alexander2008triangle}, however in this approach the sectional curvatures of both timelike and spacelike planes have to be considered. In this article, we fill a gap in the literature by proving the full equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes. As an essential tool, we establish Hessian comparison for the time separation and signed distance functions.

math.DG↗

arXiv2023

A dynamic capillarity equation with stochastic forcing on manifolds: a singular limit problem

We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold $(M,g)$. \begin{equation*}\tag{P} d \left(u_{\varepsilon,δ}-δΔ u_{\varepsilon,δ}\right) +\operatorname{div} f_{\varepsilon}(x, u_{\varepsilon,δ})\, dt =\varepsilon Δu_{\varepsilon,δ}\, dt Φ(x, u_{\varepsilon,δ})\, dW_t, \end{equation*} where $f_{\varepsilon}$ is a sequence of smooth vector fields converging in $L^p(M\times \Bbb{R})$ ($p>2$) as $\varepsilon\downarrow 0$ towards a vector field $f\in L^p(M;C^1(\Bbb{R}))$, and $W_t$ is a Wiener process defined on a filtered probability space. First, for fixed values of $\varepsilon$ and $δ$, we establish the existence and uniqueness of weak solutions to the Cauchy problem for (P). Assuming that $f$ is non-degenerate and that $\varepsilon$ and $δ$ tend to zero with $δ/\varepsilon^2$ bounded, we show that there exists a subsequence of solutions that strongly converges in $L^1_{ω,t,x}$ to a martingale solution of the following stochastic conservation law with discontinuous flux: $$ d u +\operatorname{div} f(x, u)\,dt=Φ(u)\, dW_t. $$ The proofs make use of Galerkin approximations, kinetic formulations as well as $H$-measures and new velocity averaging results for stochastic continuity equations. The analysis relies in an essential way on the use of a.s.~representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.

math.AP↗

arXiv2023

On curvature bounds in Lorentzian length spaces

We introduce several new notions of (sectional) curvature bounds for Lorentzian pre-length spaces: On the one hand, we provide convexity/concavity conditions for the (modified) time separation function, and, on the other hand, we study four-point conditions, which are suitable also for the non-intrinsic setting. Via these concepts we are able to establish (under mild assumptions) the equivalence of all previously known formulations of curvature bounds. In particular, we obtain the equivalence of causal and timelike curvature bounds as introduced in Kunzinger and S\"amann (Ann. Glob. Anal. Geom. 54(3):399-447, 2018).

math.DG↗

arXiv2022

Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

We consider the degenerate parabolic equation $$ \partial_t u +\mathrm{div} {\mathfrak f}_{\bf x}(u)=\mathrm{div}(\mathrm{div} ( A_{\bf x}(u) ) ), \ \ {\bf x} \in M, \ \ t\geq 0 $$ on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\in {\mathbb R}$, ${\bf x}\mapsto {\mathfrak f}_{\bf x}(u)$ is a vector field and ${\bf x}\mapsto A_{\bf x}(u)$ is a $(1,1)$-tensor field on $M$ such that $u\mapsto \langle A_{\bf x}(u) {\boldsymbol ξ},{\boldsymbol ξ} \rangle$, ${\boldsymbol ξ}\in T_{\bf x} M$, is non-decreasing with respect to $u$. The fact that the notion of divergence appearing in the equation depends on the metric $g$ requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem.

math.AP↗

arXiv2022

Galerkin-type methods for strictly parabolic equations on compact Riemannian manifolds

We prove existence of weak solutions to the Cauchy problem corresponding to various strictly parabolic equations on a compact Riemannian manifold $(M,g)$. This also includes strictly parabolic equations with stochastic forcing with linear diffusion. Existence is proved through a variant of the Galerkin method and can be used to construct a convergent finite element method.

math.AP↗

arXiv2022

Synthetic versus distributional lower Ricci curvature bounds

We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below $C^2$. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class $C^1$ and that the converse holds for $C^{1,1}$-metrics under an additional convergence condition on regularisations of the metric.

math.DG↗

arXiv2022

The Hawking-Penrose singularity theorem for $C^1$-Lorentzian metrics

We extend both the Hawking-Penrose Theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity $C^1$. For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the Ricci tensor now is distributional, and on the other hand, unique solvability of the geodesic equation is lost. To deal with the first issue in a consistent way, we develop a theory of tensor distributions of finite order, which also provides a framework for the recent proofs of the theorems of Hawking and of Penrose for $C^1$-metrics [7]. For the second issue, we study geodesic branching and add a further alternative to causal geodesic incompleteness to the theorem, namely a condition of maximal causal non-branching. The genericity condition is re-cast in a distributional form that applies to the current reduced regularity while still being fully compatible with the smooth and $C^{1,1}$-settings. In addition, we develop refinements of the comparison techniques used in the proof of the $C^{1,1}$-version of the theorem [8]. The necessary results from low regularity causality theory are collected in an appendix.

math-ph↗

arXiv2021

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

We study generalizations of Lorentzian warped products with one-dimensional base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

math.MG↗

arXiv2021

Null distance and convergence of Lorentzian length spaces

The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.

math.DG↗

arXiv2021

Parameter-dependent linear ordinary differential equations and topology of domains

The well-known solution theory for (systems of) linear ordinary differential equations undergoes significant changes when introducing an additional real parameter. Properties like the existence of fundamental sets of solutions or characterizations of such sets via nonvanishing Wronskians are sensitive to the topological properties of the underlying domain of the independent variable and the parameter. We give a complete characterization of the solvability of such parameter-dependent equations and systems in terms of topological properties of the domain. In addition, we also investigate this problem in the setting of Schwartz distributions.

math.CA↗

arXiv2021

A Grothendieck topos of generalized functions I: basic theory

The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a number of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is how they are still used in informal calculations in Physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, non-linear operations and the generalization of several classical theorems of calculus. Finally, we extend this category of generalized functions into a Grothendieck topos of sheaves over a concrete site. This topos hence provides a suitable framework for the study of spaces and functions with singularities. In this first paper, we present the basic theory; subsequent ones will be devoted to the resulting theory of ODE and PDE.

math.FA↗

arXiv2020

A vanishing dynamic capillarity limit equation with discontinuous flux

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the equation \begin{equation*} \begin{cases} \partial_t u_{\varepsilon,δ} +\mathrm{div} {\mathfrak f}_{\varepsilon,δ}({\bf x}, u_{\varepsilon,δ})=\varepsilon Δu_{\varepsilon,δ}+δ(\varepsilon) \partial_t Δu_{\varepsilon,δ}, \ \ {\bf x} \in M, \ \ t\geq 0 u|_{t=0}=u_0({\bf x}). \end{cases} \end{equation*} Here, ${\mathfrak f}_{\varepsilon,δ}$ and $u_0$ are smooth functions while $\varepsilon$ and $δ=δ(\varepsilon)$ are fixed constants. Assuming ${\mathfrak f}_{\varepsilon,δ} \to {\mathfrak f} \in L^p( \mathbb{R}^d\times \mathbb{R};\mathbb{R}^d)$ for some $1<p<\infty$, strongly as $\varepsilon\to 0$, we prove that, under an appropriate relationship between $\varepsilon$ and $δ(\varepsilon)$ depending on the regularity of the flux ${\mathfrak f}$, the sequence of solutions $(u_{\varepsilon,δ})$ strongly converges in $L^1_{loc}(\mathbb{R}^+\times \mathbb{R}^d)$ towards a solution to the conservation law $$ \partial_t u +\mathrm{div} {\mathfrak f}({\bf x}, u)=0. $$ The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second.

math.AP↗