Probabilistic model for exhaustion in infinite-genus curve complexes.
problem Action rigidity in infinite-genus curve complexes.
method Costa and Farber's model for random simplicial complexes.
result Probabilistic evidence for exhaustion via rigid expansions.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Rigorous model for 2-gerbes simplifies calculations in physics.
problem Constructing explicit geometric string structures.
method Defined a rigid model for bundle 2-gerbes and connections, proving equivalence to existing models.
result Chern-Simons bundle 2-gerbe can be rigidified and described via geometric string structures.
Develops a framework for stuck knots with rigid constraints and invariants.
problem Rigidity constraints on knot diagrams restrict allowable isotopies.
method Formalizes stuck crossings as rigid configurations, introduces unstick move, and constructs invariants.
result Rigidity contributes independent information even when knot type remains fixed.
Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.
problem Proving rigidity of convex polytopes in hyperbolic space.
method Spinor techniques and recent smoothing constructions of Brendle-Wang.
result Scalar curvature rigidity for parabolic convex polytopes in hyperbolic space.
New proof shows all conformal fields are Killing on specific spaces.
problem Infinitesimal conformal rigidity on Damek-Ricci spaces.
method Formulated as PDEs, analyzed locally and directly.
result Constructive proof of rigidity without global methods.
Unified rigidity theorem for Plateau surfaces in Bn.
problem Rigidity of free-boundary minimal surfaces in Bn. method Analyzing conformal free-boundary minimal immersions of Plateau model cones.
result Every conformal free-boundary minimal immersion of the flat T-cone into Bn is congruent to the flat T-cone. Rigidity theorem for spherical sectors in Riemannian manifolds.
problem Rigidity of spherical sectors in Riemannian manifolds under overdetermined conditions.
method Analyzing solutions to the inhomogeneous Helmholtz equation with constant Dirichlet and Neumann boundary conditions.
result Spherical sectors are the only solutions under given conditions.
Proves rigidity of boundaries with constant mean curvature in warped product manifolds.
problem Rigidity and compactness of boundaries with constant mean curvature in warped product manifolds.
method Distributional CMC-rigidity proof for rectifiable boundaries.
result Characterizes limits of boundaries with converging mean curvatures.
Paper proves rigidity of convex hypersurfaces in various spaces.
problem Proving the uniqueness of convex hypersurfaces in multidimensional spaces.
method Generalizing Senkin's theorem to higher dimensions and constant curvature spaces.
result Rigidity of convex hypersurfaces in En+1, n≥3. 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.
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…
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
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 MP-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.
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…
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.
Paper compares Lagrangian reduction methods for rigid body systems.
problem Modeling and reduction of rigid body systems with rotors.
method Euler-Poincaré reduction by the whole group and reduction by stages.
result Equivalence of equations and conservation laws are tracked.
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.
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.
The study shows how strictly convex domains in Euclidean spaces are rigid.
problem Understanding the rigidity of strictly convex domains in Euclidean spaces.
method Proved a rigidity theorem for smooth strictly convex domains in Euclidean spaces.
result Smooth strictly convex domains in Euclidean spaces are rigid.
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
Proves better rigidity theorems for special solitons.
problem Understanding rigidity properties of specific solitons.
method Refined point-wise estimates for mean curvature.
result Stronger rigidity results for Lagrangian and symplectic translating solitons.
Proof that specific groups are quasi-isometrically rigid.
problem Quasi-isometric rigidity of specific groups.
method Proof of quasi-isometric rigidity for Kleinian and three-manifold groups.
result Finitely generated Kleinian and three-manifold groups are quasi-isometrically rigid.
In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in Rn with n≥4. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a b…
Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.
problem Volume bounds on Alexandrov spaces' boundaries.
method Sharp volume bounds and rigidity analysis of Alexandrov spaces.
result New sharp volume bounds and classification of rigidity cases.
We define a gradient Ricci soliton to be rigid if it is a flat bundle where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci s…
Researchers prove rigidity of 2D manifolds from boundary geodesic lengths.
problem Reconstructing a Riemann surface from boundary geodesic lengths.
method Re-casting lens data as generalized Riemannian circles and solving a system of equations.
result Essentially optimal results on boundary and lens rigidity for 2D manifolds.
Quantum representations of mapping class groups are locally rigid at prime levels.
problem Locally rigid properties of quantum representations of mapping class groups.
method Proving local rigidity for Fibonacci representations of mapping class groups at prime levels.
result Local rigidity of Fibonacci representations of mapping class groups at prime levels.
Extends rigidity results to non-homogeneous manifolds.
problem Measure and topological rigidity in dynamical systems.
method From homogeneous to general manifolds.
result Measure and topological rigidity results extended.
Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
problem Spectral rigidity of hypercube graphs
method Interplay between global spectral embedding and local curvature analysis
result Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
Paper proves new rigidity results for biconservative hypersurfaces.
problem Rigidity of non-negatively curved compact biconservative hypersurfaces.
method Alternative proofs and new estimates of the squared norm of the shape operator.
result New rigidity results for biconservative hypersurfaces in space forms.
Surveying recent progress on hyperbolic 3-manifold rigidity.
problem Profinite rigidity of hyperbolic 3-manifolds.
method Review of profinite completion and rigidity of groups, evidence from other types of 3-manifolds, and existing ideas.
result Positive evidence for profinite rigidity in hyperbolic 3-manifolds.