arXiv subjects

Felix Rott

Publications and source records attributed to Felix Rott.

15 recordsLinked to original sources

arXiv2026

Total curvature and length estimates for timelike curves in Lorentzian length spaces

We introduce and study a synthetic notion of timelike total curvature for curves in Lorentzian length spaces with upper curvature bounds. In particular, we prove that our notion agrees with its smooth counterpart, and we show that timelike curves of finite total curvature are rectifiable. As the main application, we provide a sharp lower bound for the length of timelike curves solely in terms of the time separation between their endpoints and their total curvature.

math.DG↗

arXiv2026

Lorentz meets Ptolemy

We consider a Lorentzian analogue of the Ptolemy inequality and we prove that in the setting of globally hyperbolic spacetimes it is equivalent to a global timelike sectional curvature bound from above by zero. We investigate the link between the Ptolemy inequality and the hyperbolic inversion and establish some applications and rigidity properties.

math.DG↗

arXiv2026

A Splitting Theorem for non-positively curved Lorentzian spaces

We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0.

math.DG↗

arXiv2026

PDE aspects of the dynamical optimal transport in the Lorentzian setting

One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula.

math.AP↗

arXiv2025

Concavity of spacetimes

Motivated by recent breathtaking progress in the synthetic study of Lorentzian geometry, we investigate the local concavity of time separation functions on Finsler spacetimes as a Lorentzian counterpart to Busemann's convexity in metric geometry. We show that a Berwald spacetime is locally concave if and only if its flag curvature is nonnegative in timelike directions. We also give another characterization of nonnegative flag curvature by the convexity of future (or past) capsules, inspired by Krist\'aly--Kozma's result in the positive definite case. These characterizations are new even for Lorentzian manifolds.

math.DG↗

arXiv2025

Reshetnyak Majorisation and discrete upper curvature bounds for Lorentzian length spaces

We present an analogue to the Majorisation Theorem of Reshetnyak in the setting of Lorentzian length spaces with upper curvature bounds: given two future-directed timelike rectifiable curves $\alpha$ and $\beta$ with the same endpoints in a Lorentzian length space $X$, there exists a convex region in $\mathbb{L}^2(K)$ bounded by two future-directed causal curves $\bar \alpha$ and $\bar \beta$ with the same endpoints and a 1-anti-Lipschitz map from that region into $X$ such that $\bar \alpha$ and $\bar \beta$ are respectively mapped $\tau$-length-preservingly onto $\alpha$ and $\beta$. A special case of this theorem leads to an interesting characterisation of upper curvature bounds via four-point configurations which is truly suitable for a discrete setting.

math.DG↗

arXiv2024

A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes

We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is shown to satisfy certain chain and Leibniz rules, certify a locality property, and be compatible with its smooth analog. In this setup, we propose a quadraticity condition termed infinitesimal Minkowskianity, which singles out genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian in a weak form under the timelike measure contraction property. As a particular case, this extends Eschenburg's classical estimate past the timelike cut locus.

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

Characterizing intrinsic Lorentzian length spaces via $\tau$-midpoints

In metric geometry, the question of whether a distance metric is given by the length of curves can be decided via the existence of midpoints with respect to the metric $d$. We adapt a similar characterization to the setting of Lorentzian pre-length spaces. In particular, we show that a given space is strictly intrinsic provided it has $\tau$-midpoints and merely intrinsic provided it has approximate $\tau$-midpoints. Our approach is based on the null distance of C. Sormani and C. Vega.

math.MG↗

arXiv2023

A Toponogov globalisation result for Lorentzian length spaces

In the synthetic geometric setting introduced by Kunzinger and S\"amann, we present an analogue of Toponogov's Globalisation Theorem which applies to Lorentzian length spaces with lower (timelike) curvature bounds. Our approach utilises a "cat's cradle" construction akin to that which appears in several proofs in the metric setting. On the road to our main result, we also provide a lemma regarding the subdivision of triangles in spaces with a local lower curvature bound and a synthetic Lorentzian version of the Lebesgue Number Lemma. Several properties of time functions and the null distance on globally hyperbolic Lorentzian length spaces are also highlighted. We conclude by presenting several applications of our results, including versions of the Bonnet--Myers Theorem and the Splitting Theorem for Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov--Hausdorff convergence.

math.DG↗

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↗

arXiv2023

Alexandrov's Patchwork and the Bonnet--Myers Theorem for Lorentzian length spaces

We present several key results for Lorentzian pre-length spaces with global timelike curvature bounds. Most significantly, we construct a Lorentzian analogue to Alexandrov's Patchwork, thus proving that suitably nice Lorentzian pre-length spaces with local upper timelike curvature bound also satisfy a corresponding global upper bound. Additionally, for spaces with global lower bound on their timelike curvature, we provide a Bonnet--Myers style result, constraining their finite diameter. Throughout, we make the natural comparisons to the metric case, concluding with a discussion of potential applications and ongoing work.

math.DG↗

arXiv2022

The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

In this work, we prove a synthetic splitting theorem for globally hyperbolic Lorentzian length spaces with global non-negative timelike curvature containing a complete timelike line. Just like in the case of smooth spacetimes, we construct complete, timelike asymptotes which, via triangle comparison, can be shown to fit together to give timelike lines. To get a control on their behaviour, we introduce the notion of parallelity of timelike lines in the spirit of the splitting theorem for Alexandrov spaces and show that asymptotic lines are all parallel. This helps to establish a splitting of a neighbourhood of the given line. We then show that this neighbourhood has the timelike completeness property and is hence inextendible, which globalises the local result.

math.DG↗

arXiv2022

Gluing of Lorentzian length spaces and the causal ladder

We investigate the compatibility of Lorentzian amalgamation with various properties of Lorentzian pre-length spaces. In particular, we give conditions under which gluing of Lorentzian length spaces yields again a Lorentzian length space and we give criteria which preserve many steps of the causal ladder. We conclude with some thoughts on the causal properties which seem not so easily transferable.

math.DG↗

arXiv2022

Gluing constructions for Lorentzian length spaces

We introduce an analogue to the amalgamation of metric spaces into the setting of Lorentzian pre-length spaces. This provides a very general process of constructing new spaces out of old ones. The main application in this work is an analogue of the gluing theorem of Reshetnyak for CAT($k$) spaces, which roughly states that gluing is compatible with upper curvature bounds. We formulate the theorem in terms of (strongly causal) spacetimes viewed as Lorentzian length spaces.

math.DG↗