arXiv research

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 …

The paper extends completeness notions to low-regularity spacetimes.

Proves conditions for Cauchy horizons in low-regularity spacetimes.

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

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

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

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

Solves Einstein constraint equations on compact manifolds with specified boundaries.

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 …

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 …

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 timelike Ricci bounds for low regularity spacetimes using optimal transport.

Extending isometric immersions with low regularity, especially supercritical.

Low regularity spacetimes split into simpler structures.

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…

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 …

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

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

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…

Stability of timelike Ricci bounds in low-regularity spacetimes.

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.

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

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.

Analyzes Willmore flow for graphs with boundary data, proving existence and convergence.

We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.

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…

We discuss some basic concepts of semi-Riemannian geometry in low-regularity situations. In particular, we compare the settings of (linear) distributional geometry in the sense of L. Schwartz and nonlinear distributional geometry in the sense of J.F. Colombeau.

A new method for optimization in diffeological spaces using linearizations.

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

Given manifolds $M$ and $N$, with $M$ compact, we study the geometrical structure of the space of embeddings of $M$ into $N$, having less regularity than $\mathcal C^\infty$, quotiented by the group of diffeomorphisms of $M$.

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

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle $α\in (0, π)$: The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class $W_2^γ$ with $γ\in (\tfrac{3}{2}, 2]$. For …

We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has no $C^2$ conformal compactification.

We prove a Livsic type theorem for cocycles taking values in groups of diffeomorphisms of low-dimensional manifolds. The results hold without any localization assumption and in very low regularity. We also obtain a general result (in any dimension) which gives necessary and sufficient conditions to be a coboundary.

New mass definition linked to ADM mass for general metrics.

Survey on recent developments in isometric immersions using PDE techniques.

We prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space $W^{2, n/2}_{\mathrm{loc}}(M)$. We argue that this is the largest class of metrics with scalar curvature a positive a.c. measure for which the positive mass theorem may be proved by our methods.

We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance…

In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determi…

We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given po…

Synthetic proof of Gannon-Lee theorem for spacetimes.