Proves Gannon-Lee theorem for spacetimes.
problem Classical singularity theorems for spacetimes.
method Proves theorem for spacetimes, shows geodesic properties.
result Gannon-Lee theorem holds for spacetimes.
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48 results for “Regularity Theorem”
New, shorter proofs for varifolds and flows with improved decay of flatness.
problem Proving regularity theorems for varifolds and flows with bounded first variation and forcing.
method Decay of flatness via weighted monotonicity formulas and viscosity approach.
result Improved proofs with decay of flatness and characterization of blow-ups.
Establishes jet transversality for regular maps from flexible manifolds.
problem Transversality for regular maps in algebraic geometry.
method Algebraic version of Forstnerič's theorem for holomorphic maps.
result Genericity theorems for regular maps of maximal ranks.
A quaternionic version of Picard's theorem limits how many values a slice regular function can avoid.
problem How many values can a non-constant slice regular function of a quaternionic variable avoid?
method Investigates slice regular functions of quaternionic variables, extending the classical Picard theorem.
result A non-constant slice regular function of a quaternionic variable can avoid at most one value, similar to the classical Picard theorem.
The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.
problem Understanding the structure and behavior of Ricci shrinkers.
method Proving rigidity and ε-regularity theorems for Ricci shrinkers using entropy and curvature.
result Non-compact Ricci shrinkers are asymptotic to cones under certain curvature conditions.
We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity . The proof is based on regularisation techniques, combined with recent results in low regularity causality theory.
Low regularity spacetimes split into simpler structures.
problem Proving splitting theorem for metrics and weights.
method Combining elliptic techniques and line-adapted curves.
result Extends Lorentzian splitting theorem to settings.
Proves an ε-regularity theorem for Ricci flows, leading to new singularity estimates.
problem Analyzing singularity models in Fano Kähler-Ricci flows.
method Proves ε-regularity theorem and uses it to derive new estimates.
result Establishes new estimates for singularity models of Fano Kähler-Ricci flows.
Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.
problem Regularity of distance maps on geodesically complete spaces with curvature bound above.
method Define and prove regularity of distance maps as Hurewicz fibrations.
result Sphere theorem for geodesically complete CAT(1) spaces.
In this paper, we prove a Morse index theorem for the index form of regular Lagrangian system with selfadjoint boundary condition.
Proves smoothness of minimal surfaces near polyhedral boundaries.
problem Smoothness of free-boundary minimal surfaces near polyhedral domains.
method Allard-type regularity theorem for minimal surfaces in convex polyhedra.
result Minimal surfaces are graphical over a free-boundary plane if close to it.
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
Study on moduli spaces of Seiberg-Witten equations on manifolds with boundary.
problem Analyzing moduli spaces of Seiberg-Witten equations on manifolds with boundary.
method General regularity theorem, strong unique continuation principle, and gluing theorem for Dirac operators; smoothness of restriction map.
result Proves moduli spaces are Hilbert manifolds and have semi-infinite-dimensionality properties.
We establish a regularity theorem for the Harmonic - Einstein Equation. As a byproduct of the local regularity, we also have a compactness theorem on Harmonic - Einstein equation. The method is mainly the Moser iteration technique which has been used and developed by \cite{BKN89}, \cite{Tian90}, \cite{TV05a} and others…
Convergence of Ricci-limit spaces under bounded Ricci curvature and local covering geometry Imath.DG
The study extends convergence theorems for Ricci-limit spaces with bounded curvature.
problem Understanding convergence properties of Ricci-limit spaces with bounded curvature.
method Establishing -regularities and applying Fukaya's fibration theorem.
result Optimal generalization of Fukaya's fibration theorem to limit spaces.
In [Cheeger-Tian 2005], Cheeger-Tian proved an -regularity theorem for -dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in -dimensional manifolds and higher dim…
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kähler manifold. The goal of this paper is to give an -regularity theorem for the line bundle mea…
Sub-Laplacian comparison theorems on Riemannian foliations with minimal leaves and applicationsmath.DG
The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure -dimensional regular CW-complexes.
result Evenness of cells is equivalent to generalized cycle decomposition and traversability.
Consider an integral Brakke flow , , inside some ball in Euclidean space. If has small height, its measure does not deviate too much from that of a plane and if is non-empty, then Brakke's local regularity theorem yields that is actually smooth and graphical inside a smaller b…
Paper shows regularizing flow for conical Kähler-Ricci equations.
problem Regularizing property of conical Kähler-Ricci flow.
method Regularizing property of the twisted conical Kähler-Ricci flow from a positive closed current with zero Lelong number.
result Extends regularizing property to conical singularity case.
The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.
Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.
problem Optimal transport with regularizing pairs
method Local semiconcavity and future-directed timelike superdifferentials
result Derives regularizing pairs for optimal transport under general assumptions
Synthetic proof of Gannon-Lee theorem for spacetimes.
problem Proving incompleteness in globally hyperbolic spacetimes.
method Synthetic null energy condition and synthetically asymptotically regular trappedness condition.
result Generalized classical incompleteness theorem to weighted spacetimes.
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study -regular submanifolds with boundaries, prove Stokes' Theorem for them.
result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
Paper generalizes Andreev's theorem with obtuse angles.
problem Characterizing hyperbolic polyhedra with obtuse angles.
method Established discrete analog of weak solution/regularity theory.
result Generalized Andreev's Theorem to include obtuse angles.
Sharp Hölder regularity found for complex Frobenius theorem coordinates.
problem Finding optimal Hölder-Zygmund regularity for complex Frobenius theorem coordinates.
method Analyzing necessary and sufficient conditions for coordinate charts achieving the theorem's structure.
result The optimal Hölder-Zygmund regularity for coordinate charts is shown to be .
The paper proves smoothness of transition layers in the Allen-Cahn equation.
problem Proving uniform regularity for transition layers.
method Utilizes Allen-Cahn monotonicity formula, Lipschitz approximation, and blowups.
result Shows uniform regularity for transition layers converging to smooth mean curvature flows.
Generalizes Leighton's theorem to cube complexes.
problem Extending graph covering theorem to cube complexes.
method Generalizes Leighton's theorem to a family of cube complexes.
result Cube complexes have common finite covers.
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 give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…
Proves curvature bounds for close to 1 Perelman's reduced volume.
problem Curvature bounds for Ricci flow with close to 1 reduced volume.
method ε-regularity theorem for Perelman's reduced volume.
result Curvature radius cannot be too small if reduced volume is close to 1.
The paper proves a regularity theorem for Brakke flows near triple junctions.
problem Understanding the structure of triple junctions in Brakke flows.
method Establishes the ε-regularity theorem for k-dimensional Brakke flows near static, multiplicity-one triple junctions.
result The regular structure of triple junctions persists under weak mean curvature flow.
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.
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 regularity.
result Obtains isometry of higher regularity than Lipschitz.
In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in with finite Willm…
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
For any -dimensional compact spin Riemannian manifold with a given spin structure and a spinor bundle , and any compact Riemannian manifold , we show an -regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak converg…
Study on Lin-Lu-Yau curvature and diameter of amply regular graphs.
problem Lower bounds of Lin-Lu-Yau curvature in amply regular graphs.
method Application of Hall's marriage theorem and geometric proof.
result Conference graphs have positive Lin-Lu-Yau curvature.
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …
The standard approach for dealing with the ill-posedness of the training problem in machine learning and/or the reconstruction of a signal from a limited number of measurements is regularization. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augme…
The aim of this note is to present an alternative proof for an already known result relative to the solvability of the Dirichlet problem in Riemannian manifolds (see remark 0.1). In particular, we discuss the p-regularity (regularity relative to the p-laplacian) of domains of the form I = O-K, where O is a regular doma…
Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicitymath.AP
Study improves boundary smoothness for area-minimizing currents with complex boundaries.
problem Boundary regularity for area-minimizing currents with arbitrary multiplicity.
method Generalization of Allard's boundary regularity theorem.
result Derivation of structural consequences from the generalized theorem.
Alternative proof and extension of curvature estimates for minimal immersions.
problem Curvature estimates and Bernstein-type theorems for minimal immersions.
method Iteration method à la De Giorgi, ε-regularity theorem, Caccioppoli inequalities.
result Extension of Schoen--Simon--Yau and Schoen--Simon theorems to 6-dimensional stable minimal immersions.
Generalizes Frobenius theorem to quasiconformal deformations.
problem Integrability of plane fields generated by quasiconformal deformations.
method Generalization of classical Frobenius theorem to plane fields.
result A.e. involutive plane fields are integrable.
In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II of Hamilton-Jacobi equations. A…
Proves no-hair theorem for certain vacuum black holes.
problem No-hair theorem for stationary vacuum black holes.
method Proof of no-hair theorem, definition of surface gravity and angular velocity, analysis of near-horizon geometries.
result Completion of no-hair theorem proof for specified black holes.
A theorem simplifies mass-minimizing flat chains' regularity.
problem Understanding the regularity of mass-minimizing flat chains.
method Simple condition for fundamental regularity principle.
result Fundamental regularity principle holds for mass-minimizing chains.