arXiv research

The paper extends completeness notions to low-regularity spacetimes.

Proves conditions for Cauchy horizons in low-regularity spacetimes.

The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.

No trapped surfaces can form under low-regularity bounds in certain spacetimes.

Low regularity spacetimes split into simpler structures.

We study the low-regularity (in-)extendibility of spacetimes within the synthetic-geometric framework of Lorentzian length spaces developed in [KS:17]. To this end, we introduce appropriate notions of geodesics and timelike geodesic completeness and prove a general inextendibility result. Our results shed new light on …

Counterexample disproves Borde-Sorkin conjecture on causal continuity of Morse spacetimes.

In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field $φ$ on a globally hyperbolic spacetime $M$ with $C^{1,1}$ metric $g$. This first entails showing that the …

Synthetic proof of Gannon-Lee theorem for spacetimes.

We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularitie…

New relation between curvature bounds and spacetime inextendibility.

New relation between curvature bounds and spacetime inextendibility.

Stability of timelike Ricci bounds in low-regularity spacetimes.

We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity $C^{1,1}$. The proof is based on regularisation techniques, combined with recent results in low regularity causality theory.

Researchers extend the concept of metric spaces to Lorentzian spaces and prove the feasibility of their c-completion.

New proof shows FLRW spacetimes can't be extended smoothly in certain axisymmetric cases.

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open, and may differ from the corresponding sets defined via piecewise $C^1$-curves. By refining the notion of a causal…

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally…

On the Geroch-Kronheimer-Penrose future completion $IP(X)$ of a spacetime $X$, there are two frequently used topologies. We systematically examine $τ_+$, 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 maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extens…

Study on unique spacetime extensions in 1+1 dimensions with applications to weak null singularities.

The paper extends Hawking's singularity theorem to metrics with Hölder continuity and bounded curvature.

Synthetic framework for null hypersurfaces in non-smooth spacetimes.

The paper explores uniqueness and non-uniqueness of spacetime extensions in general relativity.

Geometric optics describes wave behavior near convex obstacles.

An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified concusions may arise, showing that those conclusions will fail only in special …

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $Σ\simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and the outgoing null hypersurface $\mathcal{H}$ emanating from $\partial Σ$, the time of ex…

Study proves solenoidal injectivity for tensor fields on curved manifolds with low regularity.

We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds …

Injectivity of geodesic X-ray transform on low-regularity manifolds.

Examples of area-minimizing graphs with low regularity in a specific group.

Let $\mathcal{H}$ denote the future outgoing null hypersurface emanating from a spacelike 2-sphere $S$ in a vacuum spacetime $(\mathcal{M},\mathbf{g})$. In this paper we study the so-called canonical foliation on $\mathcal{H}$ introduced by Klainerman and Nicolò and show that the corresponding geometry is controlled lo…

Smooth low-regular connections lead to smooth immersions with controlled regularity.

Solves Einstein constraint equations on compact manifolds with specified boundaries.

New bounds on ReLU networks for low-regular functions.

The paper proves a mass theorem for non-spin manifolds with low regularity curvature.

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value …

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the ge…

Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

New bounds for low-regularity Riemannian metrics defined via distributional curvature.

The study proves a transverse diameter theorem for Lorentzian foliations.

Extending isometric immersions with low regularity, especially supercritical.

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…

Stability of knots at low regularity, and symmetric critical knots for Möbius energy.

Study rigidity in low-regularity Riemannian and semi-Riemannian metrics.

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…

Paper proves DN map determination for simple surfaces with low regularity metrics.