Bing's house-like spines approximate all PL manifolds.
problem Understanding spines of PL manifolds.
method Approximation rigidity and h-principle.
result PL manifolds have spines similar to Bing's house.
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48 results for “approximation rigidity”
Study approximate marked length spectrum rigidity in non-positively curved groups.
problem Approximate rigidity of marked length spectra in non-positively curved groups.
method Compare marked length spectra of isometric actions of groups with non-positively curved features.
result Supremum of quotient of marked length spectra is approximately determined by restricted spectra.
Study shows only rotations can be approximated by Ginzburg-Landau critical points.
problem Proving not all harmonic maps can be approximated by Ginzburg-Landau critical points.
method Rigidity theorem applied to Ginzburg-Landau energy critical points.
result Only rotations can be approximated by Ginzburg-Landau critical points.
Proves rigidity for specific initial data sets under the dominant energy condition.
problem Rigidity of initial data sets with boundary and convex polytopes.
method Solution of boundary value problems for Dirac operators and approximations by manifolds with smooth boundary.
result Proves rigidity for compact smooth spin manifolds and convex polytopes under the dominant energy condition.
A flow defined by a nonsingular smooth vector field on a closed manifold is said to be parameter rigid if given any real valued smooth function on , there are a smooth funcion and a constant such that holds. We show that the parameter rigid flows on closed orientable 3-manifolds are sm…
Paper proves rigidity of CMC surfaces in curved 3-manifolds.
problem Rigidity of CMC surfaces in positive curved 3-manifolds.
method Assumptions of surface being approximately round or invariant under even symmetry, and use of Hawking mass.
result Rigidity results for stable CMC surfaces with zero Hawking mass.
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
The paper proves Weyl projective rigidity for sub-Riemannian metrics and shows genericity of such metrics.
problem Investigating the Weyl projective rigidity of sub-Riemannian metrics.
method Analytic and smooth category proofs for specific distributions with minimal order complex abnormal extremals.
result Genericity of Weyl projectively rigid sub-Riemannian metrics and distributions.
Consider a smooth manifold equipped with a bracket generating distribution . Two sub-Riemannian metrics on are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric is called rigid …
We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We a…
A new method to predict uncertainties in trained neural networks.
problem Reliability of trained neural networks outside their training domain.
method Prediction rigidities as a constrained optimization problem.
result Cheap uncertainties without modifying neural networks.
The paper explores rigidity and proximality in dynamical systems, proving new results about -algebras.
problem Understanding rigidity and proximality in dynamical systems and their algebraic counterparts.
method Analyzing crossed products of dynamical systems and their -algebras, focusing on uniform rigidity and proximality.
result Uniformly rigid systems are almost reflecting, and certain crossed products are reflecting.
Study on robustness of unsupervised representation learning in slightly misspecified settings.
problem Identify nonlinear representation learning in slightly misspecified settings.
method Formalize and investigate Independent Component Analysis (ICA) with observations generated by a mixing function close to a local isometry.
result Approximate identifiability of nonlinear ICA with almost isometric mixing functions.
In dimension , we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinker…
Locally approximating groups of homeomorphisms reveal manifold properties.
problem Understanding the structure and properties of homeomorphism groups on manifolds.
method Analyzing dense subgroups in Euclidean charts and interpreting first-order arithmetic.
result Locally approximating groups of homeomorphisms uniquely determine manifold properties.
Smoothly approximates embeddings in Lorentzian manifolds.
problem Approximating embeddings in Lorentzian manifolds.
method C^0 approximation of embeddings.
result Approximated embeddings can be made smooth.
Analytic networks with bounded coefficients can't outperform polynomial approximations.
problem Approximation limits of neural networks with analytic activation functions under coefficient constraints.
method Deterministic analysis using comparison argument and Bernstein-type estimates.
result Networks with analytic activation functions and controlled coefficients cannot outperform classical polynomial approximation rates on non-analytic targets.
The paper defines flexible domains for minimal surfaces in Euclidean spaces and explores their properties.
problem Understanding the flexibility of domains in Euclidean spaces for minimal surfaces.
method Investigates the concept of flexibility in terms of minimal surfaces contained in domains.
result Defines flexible domains and shows how they can be approximated by minimal immersions.
Real-life control tasks involve matters of various substances---rigid or soft bodies, liquid, gas---each with distinct physical behaviors. This poses challenges to traditional rigid-body physics engines. Particle-based simulators have been developed to model the dynamics of these complex scenes; however, relying on app…
New -harmonic maps of low degree are rigid under certain energy bounds.
problem Understanding the rigidity of -harmonic maps of low degree.
method Analysis of -harmonic maps and their critical points.
result Non-trivial -harmonic maps of degree zero exist with energy above .
Study automorphism groups of Artin groups, proving rigidity and classification results.
problem Understanding the structure and automorphisms of Artin groups.
method Computed automorphism groups of intersection graphs, deduced rigidity and classification results.
result Computation of outer automorphism groups and other rigidity properties.
Spinors prove rigidity for polyhedral spacetime data.
problem Rigidity of polyhedral spacetime data sets.
method Extending rigidity analysis from spacetime positive mass theorem.
result Dihedral rigidity connects mass theorem, trapped surfaces.
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some r…
Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
problem Rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
method Survey and observation on Cheeger-Yau inequality on RCD spaces.
result Observations on the Cheeger-Yau inequality and its applications.
In this article, we discuss the local rigidity of Clifford-Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. In fact, we introduce a splitting of the local rigidity: vertical rigidity and horizontal rigidity. By using this splitting, we refine some existing results about the local rigidit…
A real valued function of one variable is called a metric transform if for every metric space the composition is also a metric on . We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms such that the trans…
Study shows critical width for rigidity of equatorial zones on spheres.
problem Mean curvature rigidity of equatorial zones on spheres.
method Used tangency principle and trap-slice lemma for strong rigidity, and constructed nontrivial perturbations using Delaunay surfaces for non-rigidity.
result Critical width exists for rigidity, beyond which zones are non-rigid.
Projective structures are mostly rigid at the boundary but some are not.
problem Boundary rigidity of projective structures.
method Investigation of projective structures on manifolds with boundary.
result Existence of non-rigid projective structures and characterization of them.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
Non-rigidity of hyperbolic manifold under scalar curvature constraints.
problem Non-rigidity of hyperbolic manifold under scalar curvature constraints.
method Compactly supported deformations, topological constraints.
result Non-rigidity under scalar curvature constraints, rigidity under topological constraints.
We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?
Study scattering rigidity on stationary manifolds using geodesics.
problem Scattering rigidity on standard stationary manifolds.
method Use Hamiltonian reduction to relate to -systems.
result New rigidity results for stationary manifolds.
Study rigid classes on hyperkahler manifolds, showing general ones are rigid.
problem Characterize rigid classes on compact hyperkahler manifolds.
method Analyze eigenvectors of hyperbolic automorphisms and use BBF form.
result General parabolic classes on hyperkahler manifolds are rigid.
In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by isometric embeddings. This statement clearly cannot be true for embeddings in general, due to the classi…
We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions u…
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains…
We prove the developability and regularity of isometric immersions of -dimensional domains into . As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the strong norm, provided the domain is and convex. Both results fail to …
The flip graph and arc complex of a surface are shown to have finite rigidity.
problem Finite rigidity of flip graph and arc complex for surfaces.
method Embedding the flip graph in the arc complex and leveraging finite rigidity of the flip graph.
result Finite rigidity of the flip graph implies finite rigidity of the arc complex.
Every noncompact surface has a 3-rigid triangulation.
problem Classifying noncompact surfaces and proving their rigidity.
method Triangulation and minimally rigid structures.
result Every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid.
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results…
Rigidity proven for a specific type of solitons with harmonic curvature.
problem Proving rigidity of a specific class of solitons.
method Proof of rigidity for compact three-dimensional Heterotic solitons with vanishing torsion and harmonic curvature.
result Compact three-dimensional Heterotic solitons with vanishing torsion and harmonic curvature are rigid.
A rigid submanifold result in contact geometry.
problem Rigidity of coisotropic submanifolds in contact geometry.
method Study coisotropic deformations and characteristic foliations.
result Compact regular coisotropic submanifolds are rigid among nearby ones.
Investigates conditions for non-rigidity in extremal metrics involving scalar curvature.
problem Rigidity of extremal metrics involving scalar curvature.
method Analyzes sufficient conditions for non-rigidity and provides examples.
result Provides sufficient conditions for metrics not to be rigid.
Totally geodesic subvarieties in moduli space are locally rigid.
problem Understanding rigidity of subvarieties in moduli space.
method General rigidity result for orbifold maps to moduli space.
result Covering constructions and totally geodesic subvarieties are locally rigid.
New rigidity results for complex and quaternionic moment-angle manifolds.
problem Equivariant topological rigidity of complex and quaternionic moment-angle manifolds.
method Reduction to equivariant rigidity of quasitoric (or quoric) quotients and principal bundles.
result Full equivariant rigidity for manifolds with four-dimensional quoric quotients and primary rigidity for higher dimensions.
New spectral conditions ensure graph rigidity and global rigidity in the Euclidean plane.
problem Ensuring graph rigidity and global rigidity in the Euclidean plane.
method Improving algebraic connectivity bounds for graph rigidity and global rigidity.
result Every 6-connected graph is rigid and globally rigid if its algebraic connectivity exceeds specific thresholds.
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.