Study proves solenoidal injectivity for tensor fields on curved manifolds with low regularity.
problem Injectivity for tensor fields on negatively curved manifolds with low regularity metrics.
method Pestov energy estimates for transport equation on non-smooth unit sphere bundle, keeping track of regularity, and using functions with more vertical than horizontal regularity.
result Proves solenoidal injectivity for tensor fields on simple Riemannian manifolds with C1,1 metrics and non-positive sectional curvature. New bounds on ReLU networks for low-regular functions.
problem Bounding approximation error for ReLU networks on low-regular functions.
method Complexity analysis of Fourier features residual networks to ReLU networks.
result Approximation error bound proportional to target function norm and inversely proportional to network width and depth.
We study ray transforms on spherically symmetric manifolds with a piecewise C1,1 metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of L2 functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds …
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
Injectivity of geodesic X-ray transform on low-regularity manifolds.
problem Injectivity of geodesic X-ray transform on manifolds with low regularity.
method Calculus of differential and curvature operators on non-smooth structures.
result Injectivity of geodesic X-ray transform on simple Riemannian manifolds with C1,1-regularity. 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 C1,1 metric g. This first entails showing that the …
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≃B(0,1)⊂R3 and the outgoing null hypersurface H emanating from ∂Σ, the time of ex…
The paper extends completeness notions to low-regularity spacetimes.
problem Defining completeness conditions for spacetimes with low-regularity metrics.
method Extending Beem's completeness notions to Lorentzian length spaces and proving relationships between them.
result Equivalence of completeness conditions for globally hyperbolic C1-spacetimes under certain conditions. Counterexample disproves Borde-Sorkin conjecture on causal continuity of Morse spacetimes.
problem Disproving the Borde-Sorkin conjecture on causal continuity of Morse spacetimes.
method Provided a counterexample with low regularity causal structure and causal bubbling.
result Borde-Sorkin conjecture does not hold for Morse spacetimes with large anisotropy.
Proves conditions for Cauchy horizons in low-regularity spacetimes.
problem Conditions for the existence of Cauchy horizons in spacetimes with low regularity.
method Analyzes the relationship between complete Cauchy hypersurfaces, almost closed causal curves, and points at infinity.
result Wald's conjecture reformulated as a PDE problem about Cauchy horizons.
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.
No trapped surfaces can form under low-regularity bounds in certain spacetimes.
problem Existence of trapped surfaces in low regularity solutions to Einstein's equations.
method Analyzing the initial data in Besov B2,13/2 norm and extending to H3/2 smallness. result No trapped surfaces can exist initially when the Cauchy data are close to Minkowski spacetime data.
Paper proves DN map determination for simple surfaces with low regularity metrics.
problem Determining DN map from scattering relation for surfaces with low regularity metrics.
method Modified technical results and used microlocal analysis for metrics with finite regularity.
result Scattering relation determines DN map for C17 surfaces, and for C1,1 metrics using Lipschitz distance function. Examples of area-minimizing graphs with low regularity in a specific group.
problem Finding area-minimizing graphs with low regularity in a sub-Finsler Heisenberg group.
method Providing examples of entire area-minimizing horizontal graphs with prescribed singular sets.
result Examples of area-minimizing graphs that are locally Lipschitz but not necessarily smoother.
Smooth low-regular connections lead to smooth immersions with controlled regularity.
problem Smoothability of Lp-connections and existence of isometric immersions with low regularity. method Adapting S. Mardare's work on surface theory, using Hodge decomposition and fixed point theorems.
result Low-regular connections can be approximated by smooth connections of the same curvature.
We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle α∈(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 W2γ with γ∈(23,2]. For …
Solves Einstein constraint equations on compact manifolds with specified boundaries.
problem Solving Einstein constraint equations with specified boundaries.
method Studies conformal constraint equations with low regularity assumptions.
result 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 …
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…
The paper proves a mass theorem for non-spin manifolds with low regularity curvature.
problem Establishing a mass theorem for non-spin manifolds with low regularity curvature.
method Smooth approximations of the metric, Sobolev version of Friedrichs' Lemma, comparison theory of RCD-spaces, rigidity theorem for compact manifolds.
result Asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass.
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 …
The paper extends Hawking's singularity theorem to metrics with Hölder continuity and bounded curvature.
problem Proving singularity theorems for metrics with low regularity.
method Combining elliptic RT-equations for metric regularisation and manifold convolution for curvature refinement.
result Establishes globally hyperbolic and timelike incompleteness for metrics with Hölder continuity and bounded curvature.
We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subse…
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>23. 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.
problem Establishing curvature bounds for Riemannian metrics of low regularity.
method Introducing a distributional version of sectional curvature for C1 and C0 metrics. result New bounds for low-regularity metrics recover classical bounds in Alexandrov spaces.
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and…
The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.
problem Proving timelike Ricci bounds for spacetimes with low regularity.
method Using optimal transport to prove timelike measure-contraction property.
result Timelike curvature-dimension condition holds for C1,1 metrics. Extending isometric immersions with low regularity, especially supercritical.
problem Finding isometric immersions with low regularity in Euclidean space.
method Utilising Uhlenbeck gauges and compensated compactness theory.
result Existence of isometric immersions with low regularity, including supercritical cases.
Low regularity spacetimes split into simpler structures.
problem Proving splitting theorem for C1 metrics and weights. method Combining elliptic techniques and line-adapted curves.
result Extends Lorentzian splitting theorem to C1 settings. 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.
problem Stability of knot equivalence at low regularity.
method Localized Gromov distortion and Hausdorff-distance criteria.
result Compactness theorem for knot equivalence classes and existence of symmetric critical knots for Möbius energy.
Study rigidity in low-regularity Riemannian and semi-Riemannian metrics.
problem Rigidity problems for low-regularity metrics.
method Proves Cheeger-Gromoll splitting theorem and flatness criterion for semi-Riemannian metrics of C1 regularity. result Obtains isometry of higher regularity than Lipschitz.
We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the BV fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regul…
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 C1-curves. By refining the notion of a causal…
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
Regularity results for geodesic X-ray transform on nonsmooth manifolds
problem Geodesic X-ray transform on nonsmooth simple manifolds
method Symbol smoothing arguments and pseudodifferential operators with low regularity symbols
result Improved injectivity results for Lp functions 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…
We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity C1,1. 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.
problem Willmore flow of graphs with boundary conditions over bounded domains.
method Developed low-regularity theory, reformulated graphical equation, used time-weighted parabolic Hölder spaces.
result Proved short-time and global existence for initial data in C1+α(Ω) and Lipschitz, with exponential convergence. Deep adaptive sampling improves surrogate modeling for complex systems.
problem Statistical errors in random sampling for high-dimensional problems.
method DAS^2 method, using deep generative models to refine training sets.
result Reduces statistical errors in approximating solutions for low-regularity problems.
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.
problem Optimization in spaces with low regularity.
method Generalizing linearization to diffeological spaces and constructing smooth paths.
result Achieving weak convergence to minima or critical values in diffeological spaces.
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 C∞, 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.
problem Extending the concept of metric spaces to Lorentzian spaces and proving their c-completion.
method Revisiting Lorentzian metric spaces, constructing c-completion, proving feasibility and endowing with Lorentzian metric space structure.
result The c-completion of Lorentzian metric spaces is feasible and well-suited, completing the original space in a precise sense.
We prove the existence of a C1,1 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 C2 conformal compactification.
Graph diffusion processes approximate manifold heat semigroups using graph transition matrices.
problem Approximating manifold heat semigroups from graph data under low regularity conditions.
method Iterating graph transition matrix P to approximate Qt=etΔ, bounding error in ∞-norm. result Convergence rates O(N−2/(d+6)) for manifold heat semigroup approximation, valid for in-sample and out-of-sample. 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.
problem Defining mass for metrics with low regularity.
method Using isocapacitary inequality to define total mass.
result Inequality between new mass and ADM mass proved.