Proves existence of low regularity Einstein metrics on the ball.
problem Existence of low regularity conformally compact Einstein metrics.
method Proves existence of C1,1 conformally compact Einstein metric with specific curvature decay. result Existence of C1,1 conformally compact Einstein metric with asymptotic curvature decay. Study geodesics in low regularity metrics, proving existence and uniqueness.
problem Existence and uniqueness of geodesics in low regularity metrics.
method Analyzes Riemannian and Lorentzian manifolds with low regularity metrics, proving existence and uniqueness of solutions.
result Generalizes recent results on existence, regularity, and uniqueness of solutions for geodesics in low regularity metrics.
Study of ray transforms on spherically symmetric manifolds with low regularity metrics.
problem Injectivity of X-ray transforms and broken ray transforms on spherically symmetric manifolds with low regularity metrics.
method Introduced and studied generalized Abel transforms to handle low regularity settings.
result Injectivity results for X-ray and broken ray transforms on spherically symmetric manifolds with C1,1 metrics. Essential self-adjointness proven for powers of first-order differential operators on non-compact manifolds with low-regularity metrics.
problem Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics.
method Demonstrates the equivalence between essential self-adjointness and a negligible boundary property for first-order differential operators with locally bounded measurable coefficients. For higher regularity coefficients, essential self-adjointness of higher powers is shown under the same condition.
result Essential self-adjointness of first-order differential operators and their higher powers proven under specific conditions on non-compact manifolds with low-regularity metrics.
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.
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.
Extends singularity theorems to low regularity metrics.
problem Proving singularity theorems for metrics with low regularity.
method Careful estimates of curvature and stability properties of geodesics.
result Complete proof of Hawking and Penrose singularity theorems for C1-Lorentzian metrics. 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…
Develops Green operators for quantum fields on low-regularity spacetimes.
problem Describes quantum fields on low-regularity spacetimes with C1,1 metrics. method Shows well-posedness of wave equation, constructs Green operators, defines symplectic form.
result Provides a locally covariant description of quantum fields.
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.
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.
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. 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. 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. 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. 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. 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.
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 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…
The Myers-Steenrod theorem is extended to Finsler manifolds with low regularity.
problem Extending the Myers-Steenrod theorem to Finsler manifolds with minimal regularity.
method Reduction of Finslerian problems to Riemannian ones using the Binet-Legendre metric.
result Isometries between Ck,α-smooth Finsler metrics are diffeomorphisms of class Ck+1,α. 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 prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space Wloc2,n/2(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.
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.
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.
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.
Paper studies Einstein vacuum equations with low regularity data.
problem Einstein vacuum equations with low regularity initial data.
method Combines Klainerman-Szeftel-Rodnianski curvature theorem, Czimek's extension procedure, and global elliptic estimates.
result Time of existence controlled by low regularity bounds on curvature in L2. 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.
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 findings show non-open chronological futures in low regularity spacetimes.
problem Breakdown of Lorentzian causality theory in low regularity spacetimes.
method Refined notion of causal bubble and analysis of locally Lipschitz curves.
result Chronological futures may be non-open and differ from those defined via piecewise C1-curves. 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.
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 mass theorems for nonnegative scalar curvature metrics.
problem Proving mass theorems for metrics with nonnegative scalar curvature.
method New local inverse mean curvature flow with quantitative stability.
result Existence of isoperimetric sets in low regularity metrics.
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. 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.
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…
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.
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.
New method finds smooth isometric immersions for low regularity metrics, achieving full flexibility.
problem Finding smooth isometric immersions for metrics with low Hölder regularity.
method Techniques of convex integration to find isometric immersions of low regularity.
result Achieves full flexibility, reaching C1,1− for Cr,β metrics. 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.
Maximal Schwarzschild spacetime is inextendible with continuous metric.
problem Inextendibility of Schwarzschild spacetime with continuous metric.
method Introduced spacelike diameter to capture curvature singularity obstruction.
result Proved inextendibility of Schwarzschild spacetime with continuous metric.
Study on extendibility of spacetimes using synthetic geometry.
problem Analyzing extendibility of spacetimes with low regularity.
method Introducing geodesics and proving inextendibility results in Lorentzian length spaces.
result Relate low-regularity inextendibility to curvature blow-up.
We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space Wloc2,n/2 for manifolds of dimension less than or equal to 7 or spin-manifolds of any dimension. More generally, we give a (negative) lower bound on the ADM mass of metrics for which the scalar curvat…
The study shows that positive scalar curvature 4-manifolds can be desingularized.
problem Proving the non-existence of singular positive scalar curvature metrics.
method Explicit constructions from scalar-flat Kähler ALE surfaces and desingularization process.
result Desingularization of positive scalar curvature 4-manifolds is possible.
Study of null hypersurface foliation in low regularity spacetimes.
problem Control the geometry of null hypersurfaces in low regularity spacetimes.
method Generalised methods of Klainerman and Rodnianski, Alexakis, Shao and Wang.
result Locally uniformly bounded ingoing and outgoing null expansions.
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.
We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Ise…
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.
Survey shows degenerate elliptic equations have analytic properties despite low regularity.
problem Understanding analytic properties of degenerate elliptic equations.
method Explains why solutions of a specific degenerate elliptic equation are analytic.
result Solutions of a specific degenerate elliptic equation are analytic despite low regularity.