Study disproves conjecture about metric completion of curve spaces.
problem Completeness properties of spaces of immersed curves with reparametrization-invariant metrics.
method Examined Sobolev-type metrics on real-valued immersed curves, demonstrating multiple distinct limit points.
result Metric completion of spaces of immersed open curves includes multiple distinct limit points, not a single point as previously conjectured.
Completeness of Sobolev metrics on curve spaces proven.
problem Proving completeness of Sobolev metrics on curve spaces.
method Analyzing Sobolev metrics with nonconstant coefficients on curve spaces.
result Necessary and sufficient conditions for metric completeness provided.
Alternative proof for non-existence of complete curves in differential strata.
problem Non-existence of complete algebraic curves in strata of holomorphic differentials.
method Using positivity of divisor classes on moduli spaces of curves.
result Alternative proof confirming Gendron's result on non-existence.
Study on completeness of Sobolev metrics on manifold-valued curves.
problem Completeness of Sobolev metrics on spaces of manifold-valued curves.
method Analysis of reparametrization invariant Sobolev metrics of order n≥2. result Sobolev immersions are metrically and geodesically complete for several important cases of metrics.
We prove that any convex domain of C^2 carries properly embedded complete complex curves. In particular, we exhibit the first examples of complete bounded embedded complex curves in C^2
We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order n≥2 are metrically complete on the space In(S1,Rd) of Sobolev immersions of the same regularity and that any two curves i…
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.
Study on Brownian motion on discrete curve spaces, proving stochastic completeness.
problem Analyzing Brownian motion on spaces of discrete curves.
method Introduced and studied Brownian motion on spaces of discrete regular curves with Sobolev-type metrics.
result All geodesically complete spaces of discrete regular curves are stochastically complete.
Study shows how brain completes missing parts of curves and constructs surfaces of negative curvature.
problem How the brain completes missing parts of curves and constructs surfaces of negative curvature.
method Solving variational problems to find sub-Riemannian geodesics and constructing surfaces of constant negative curvature.
result There is a one-to-one correspondence between sub-Riemannian geodesics used by the brain and rotational surfaces of constant negative curvature.
Study preserves planar and graphical properties of curves under elastic flow.
problem Maintaining planar and graphical properties of non-compact curves under elastic flow.
method Extended recent work on adapted elastic energy to derive thresholds for planar and graphical embeddedness.
result Derived new Li--Yau type inequality for complete planar curves.
Constructs negatively curved complete intersections in complex manifolds.
problem Creating compact negatively curved complete intersections.
method Using Donaldson-Auroux theory to construct and prove existence.
result Existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature.
Characterizes Coxeter groups with specific boundary shapes.
problem Identifying Coxeter groups with Sierpiński or Menger curve boundaries.
method Combining results from the literature on Gromov boundaries and Coxeter groups.
result Complete characterizations of hyperbolic Coxeter groups with Sierpiński or Menger curve boundaries.
Study of monodromy and vanishing cycles for complete intersection curves.
problem Computing topological monodromy of complete intersection curves.
method Innovative tools for studying monodromy of tensor products of very ample line bundles, induction on multi-degree.
result Answer given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.
problem Investigating geometric properties of immersed curves with fractional Sobolev metrics.
method Analyzing Riemannian metrics on spaces of immersed curves, proving completeness and geodesic properties.
result Fractional Sobolev metrics are geodesically complete for q>3/2. Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the ∞-category of curved Lie algebras using homotopy theory of algebras over a complete operad. result Equivalence of ∞-categories between Lie algebroids and certain kinds of curved Lie algebras. Totally geodesic dual leaves on curved manifolds are also curved.
problem Characterizing dual leaves of nonnegatively curved polar manifolds.
method Proving dual leaves are totally geodesic and closed, and inducing a Riemannian submersion.
result Dual leaves of nonnegatively curved polar manifolds are themselves nonnegatively curved and totally geodesic.
Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.
problem Classifying geodesic curves on the Heisenberg group.
method Completely integrable Hamiltonian system, classification of geodesic curves.
result Complete classification of geodesic curves on the Heisenberg group.
Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the s…
The ball in complex 2-space can contain curves of any shape.
problem Embedding complex curves of arbitrary topology in the ball of C2. method Proving existence of curves with any given topological type.
result Complete embedded complex curves of any topological type exist in the ball of C2. In this paper we study the general affine geometry of curves in affine space A2. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…
A new Riemannian metric on curve spaces is complete and smooth.
problem Defining a complete metric on the space of embedded curves.
method Proposed a new Riemannian metric and proved its completeness.
result The proposed metric is complete in multiple senses.
The paper examines affine properties of differential strata on curves.
problem Characterizing affine structures in strata of differentials.
method Analyzing affine varieties and Teichmüller dynamics.
result Affine strata do not contain complete curves with certain pole orders.
Paper computes braid monodromy for special curves using a new method.
problem Computing invariants of completely reducible n-gonal curves. method Rectangular braid diagram method and Burau representations.
result Alexander polynomial of curve complements computed successfully.
In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface S of negative Euler characteris…
Formula for winding numbers on non-null-homotopic curves on surfaces.
problem Determining winding numbers for non-null-homotopic curves on surfaces.
method Generalized a Whitney-type formula for winding numbers of non-null-homotopic curves on aspherical surfaces.
result Formula for winding numbers on non-null-homotopic curves on aspherical surfaces.
Proves conjecture about foliations on curved spaces.
problem Completeness of dual foliations on curved spaces.
method Analyzes Riemannian foliations on nonnegatively curved symmetric spaces.
result Foliations split into trivial and single dual leaf foliations.
We give a complete description of all order 1 invariants of planar curves.
No primitive Teichmüller curves found in Prym(2,2).
problem Existence of primitive Teichmüller curves in Prym(2,2).
method Completed work by Lanneau and Möller.
result No primitive Teichmüller curves in Prym(2,2).
Complete conjecture on rational curves on K3 surfaces.
problem Existence of infinitely many rational curves on K3 surfaces.
method Two new techniques: regeneration and marked point trick.
result Existence of integral curves of unbounded degree for any projective K3 surface.
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
Abstract: Defines differential equations in tangent categories, providing conditions for completeness and new perspectives.
problem Defining and working with differential equations in abstract tangent categories.
method Introduces curve objects and dynamical systems, providing conditions for completeness and exploring exponential maps.
result Provides abstract conditions for dynamical systems to be complete and introduces differential exponential rig.
Study shows fundamental groups of certain curved spaces are simple.
problem Understanding fundamental groups of curved spaces with specific growth conditions.
method Proved using nonnegative Ricci curvature and volume growth conditions.
result Fundamental groups are trivial or finite for these spaces.
No Shimura-Teichmüller curves found in genus 5.
problem Classifying Shimura-Teichmüller curves in genus 5.
method Utilized the equivalence of Shimura-Teichmüller curves to having completely degenerate Kontsevich-Zorich spectrum, and implemented a computer search to exclude remaining cases.
result No Shimura-Teichmüller curves exist in genus 5.
Classifies real algebraic curves on a quadric ellipsoid of specific degree.
problem Classifying real algebraic curves of bidegree (5,5) on the quadric ellipsoid.
method Reduction to curves in the second Hirzebruch surface, combining classical construction methods on toric surfaces.
result Previously known restrictions form a complete system for this bidegree.
In this paper, we completely classify the magnetic curves (also N-magnetic curves with constant curvature) in a Galilean 3-space associated to a Killing vector field.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
We give a complete description of all order 1 invariants of spherical curves. We also identify the subspaces of all J-invariants and S-invariants, and present two equalities satisfied by any spherical curve.
New algebraic theory classifies symplectic curves in complex projective space.
problem Classifying symplectic curves with specific singularities.
method Developed a novel algebraic theory of positive braids and conjugacy classes in the braid group.
result Established a complete classification of isotopy classes of degree three symplectic curves with An-singularities. Let Out(Fn) be the outer automorphism group of the free group Fn. It acts properly on the outer space Xn of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise s…
Study magnetic curves in Sasakian manifolds, classifying and parametrizing them.
problem Classify and parameterize pseudo-Hermitian magnetic curves in Sasakian manifolds.
method Define and classify pseudo-Hermitian magnetic curves, construct parametrizations.
result Complete classification theorem for pseudo-Hermitian magnetic curves in Sasakian manifolds.
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.
For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to t…
Algorithm detects free products in disk mapping class groups.
problem Detecting free products in mapping class groups of punctured disks.
method Algorithm based on Dynnikov coordinates to verify completeness and reveal free product structure.
result Algorithm determines exact structure of free products generated by Dehn twists.
Study finds minimum lengths of curves on a one-holed torus.
problem Minimizing the geodesic length of curves on a one-holed torus.
method Explicitly found minima and minimum points of geodesic length functions for a family of curves.
result Concrete examples provided for minimizing geodesic length on hyperbolic surfaces.
We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane…
Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negative…
This paper solves a question about curves on toric surfaces.
problem Which curves can be vanishing cycles for degenerations?
method Reformulated as a mapping class group problem and determined the monodromy group.
result A complete answer to the question of which curves can be vanishing cycles.