Finite rigid sets found in surface curve complexes.
problem Finding rigid sets in surface curve complexes.
method Incidence-preserving maps to find rigid subcomplexes.
result Finite rigid subcomplexes identified in surface curve complexes.
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48 results for “rigid curves”
Finite rigid sets found in complex of curves for surfaces.
problem Finding finite rigid sets in curve complexes of surfaces.
method Exhaustion by finite rigid sets proved for surfaces of finite type and genus ≥3.
result Finite rigid sets exist in the non-separating curve complex of surfaces.
Sharp estimate shown to be rigid on curved surfaces.
problem Sharp Bezout estimate on nonnegatively curved Riemann surfaces.
method General three circle theorem applied.
result Rigidity of the sharp Bezout estimate.
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.
For an orientable surface of finite topological type with genus , we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of . The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a …
The study proves rigidity for mixed Hodge structures and applies to curve families.
problem Rigidity of period maps for mixed Hodge structures.
method Holomorphic bisectional curvature approach.
result Establishes rigidity in various cases, including curve families.
The study constructs balanced and rigid curves on specific types of hypersurfaces and complete intersections.
problem Constructing balanced and rigid curves on Calabi-Yau and general-type complete intersections.
method Balanced and rigid curves are constructed using specific hypersurfaces and complete intersections.
result Rigid curves of various genera and balanced rational curves of high degrees are constructed.
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.
A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected non-orientable surfaces of genus with holes …
This study exhausts curve graphs of low-genus surfaces.
problem Exhausting curve graphs of low-genus surfaces.
method Constructing finite subgraphs and using rigid expansions.
result Graph morphisms and endomorphisms are automorphisms and induced by homeomorphisms.
Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
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.
New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.
problem Identifying surfaces by their geodesic lengths.
method Analyzes metrics on simple, thick negatively curved two-dimensional P-manifolds.
result Piecewise negatively curved Riemannian metrics on simple, thick two-dimensional P-manifolds are uniquely determined by their geodesic lengths.
The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.
problem Determining metrics on negatively curved manifolds using spectral data.
method Analyzing conjugacy classes and marked length spectra.
result Sparse sets exist that uniquely determine metrics on negatively curved manifolds.
Finite rigid sets found in sphere complexes for some but not all cases.
problem Characterizing finite rigid sets in sphere complexes.
method Analyzing locally injective maps and automorphisms.
result Finite rigid sets exist for but not for .
Paper studies rigidity of translation surfaces in 3D sphere using quaternionic product.
problem Rigidity of translation surfaces in .
method Introduced an associated frame for curves in ; described local geometry; used curvature and torsion of generating curves.
result Rigidity results for minimal and constant mean curvature surfaces in .
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.
This paper exhausts curve complexes on non-orientable surfaces.
problem Proving exhaustion of curve complexes on non-orientable surfaces.
method Proving exhaustion via rigid expansions and graph endomorphisms.
result Any graph endomorphism of curve complexes whose restriction to a finite rigid set is injective is induced by a homeomorphism.
We prove that any properly oriented isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable…
Let be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex by a sequence of finite rigid sets.
New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.
problem Rigidity of sharp spectral gap in nonnegatively curved spaces.
method Mixing Sobolev theory and singular 1D-localization.
result Rigidity of in compact spaces.
In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…
The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.
problem Characterizing hyperbolic surfaces by their simple length spectra.
method Proving rigidity of length identities over Teichmüller spaces.
result Simple length spectra can be used as moduli for generic hyperbolic surfaces.
Improved bounds on curve filling areas in Banach spaces, leading to rigidity of Pu's inequality.
problem Improving bounds on curve filling areas in non-geodesic Banach spaces.
method Improved bounds on curve filling areas in Banach spaces.
result Rigidity of Pu's classical systolic inequality.
This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved me…
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.
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
The paper characterizes simple closed curves on surfaces using profinite rigidity.
problem Characterizing simple closed curves on surfaces using profinite rigidity.
method Proving that elements with the same images under all finite groups are simple closed curves.
result The set of simple closed curves is closed in the profinite topology of the surface group.
Global rigidity theorem for curve to abelian variety maps.
problem Characterizing maps between moduli spaces of curves and abelian varieties.
method Analyzing nonconstant holomorphic maps between moduli spaces.
result Holomorphic maps between specific moduli spaces are rigid, with only one possible form.
New rigidity result for hyperbolic surfaces based on curve lengths.
problem Determining hyperbolic metrics on surfaces from curve lengths.
method Investigating oriented graphs on curve complexes and Dehn quasi-homothetic functions.
result Knowing which curve is longer suffices to determine the hyperbolic metric on a surface.
New method proves length spectrum rigidity in various geometric settings.
problem Length spectrum rigidity in geometric settings.
method Combination of dynamical systems and geometric group theory.
result Provides concise proofs and extends classical results.
New counterexample shows curved surfaces can deform geodesics without diffeomorphism.
problem Can curved surfaces deform geodesics without changing their lengths?
method Constructs a perturbed surface with longer geodesics but no contracting diffeomorphism.
result No diffeomorphism can contract all tangent vectors on a surface with longer geodesics.
New method for curve comparison using iterated integrals and moving frames.
problem Comparing curves robustly to noise and transformations.
method Moving frame method paired with log-signature transform.
result Algorithmic construction of invariants for curve equivalence under rigid motions.
A new FFT-based method for fast rigid alignment of 2D closed curves.
problem Rigid alignment of 2D closed curves with application to shape analysis.
method FFT-based algorithm for optimal rigid alignment of closed curves with O(N log N) complexity.
result Order of magnitude speed-up in curve alignment compared to previous methods.
We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the vertical tangent bundle is topologically rigid.
Study automorphisms on procongruence curve and pants complexes.
problem Understanding automorphism groups of procongruence curve and pants complexes.
method Action on procongruence mapping class group and rigidity theorem for pants complex.
result Prove rigidity theorem for procongruence completion of pants complex.
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.
Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.
problem Stochastic dynamics of rigid inclusions on curved surfaces.
method Cartan's method of moving frames, Hamiltonian equations, intrinsic Langevin equations, Fokker-Planck equation.
result Extracted overdamped equations for accurate simulations of diffusion processes.
New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.
problem Proving rigidity of minimal hypersurfaces in curved 4-manifolds.
method Combining nonnegative 2-intermediate Ricci curvature and strict positivity of scalar curvature, extending Chodosh-Li-Stryker method.
result Rigidity of two-sided free boundary stable minimal hypersurfaces in 4-manifolds with bounded geometry and weakly convex boundary.
We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy …
New findings on stable minimal hypersurfaces in curved 4-manifolds.
problem Nonexistence of complete stable minimal hypersurfaces in positively curved 4-manifolds.
method Combination of non-negative sectional curvature and strict positivity of scalar curvature.
result Rigidity of complete stable minimal hypersurfaces in 4-manifolds with positive 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.
We study the horizontally regular curves in the Heisenberg groups . We show the fundamental theorem of curves in and define the concept of the orders for horizontally regular curves. We also show that the curve is of order if and only if lies in but not in up to a Heis…
Study on choosing points on cubic curves, answering some questions about their flexibility.
problem Determining if algebraic structures can continuously choose points on cubic plane curves.
method Analyzing the flex points and sextatic points of cubic plane curves.
result Affirmative answer for and 18, negative for infinitely many .
The Manhattan curve connects metrics of hyperbolic groups, showing rigidity.
problem Understanding the relationship between different metrics on hyperbolic groups.
method Ergodic theory of topological flows and analysis of Patterson-Sullivan measures.
result The Manhattan curve is a straight line if and only if metrics are roughly similar.
In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension , with non-empty totally geodesic boundary. More precisely, if are any two such manifolds, we show that (1) is homeomorphic to $\partial ^\infty…
The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.
problem Investigating rigidity and deformability of pseudoholomorphic curves in .
method Analyzing moduli space of minimal surfaces isometric to pseudoholomorphic curves.
result Describes the moduli space of noncongruent minimal surfaces isometric to pseudoholomorphic curves.
Curve complexes are rigidmath.GT
Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.