arXiv research

New method measures entanglement of open and closed curves.

Defines Vassiliev complexity measures for open and closed curves in 3D space.

New invariants measure entanglement of open curves in 3D space.

Novel Jones polynomial for open curves in 3D space.

Solves open problems on curved projective varieties.

Knoto-ID studies the entanglement of open protein chains without closing them.

We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of const…

Simple curves enclose two small disks if they're wide and bend moderately.

Curve diffusion flow straightens curves with endpoints on intersecting lines.

The study examines elastic curves with self-intersections and their properties.

Study disproves conjecture about metric completion of curve spaces.

This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.

New findings show non-open chronological futures in low regularity spacetimes.

Survey on rational curves on complex surfaces, highlighting different approaches.

New method identifies knots in protein chains using virtual knots.

The study shows that ergodic measures are not generic on non-positively curved manifolds.

Dubrovin duality connects two F-manifolds on the universal curve.

Study shows open manifolds can have non-homeomorphic souls with positive curvature.

Study on completeness of Sobolev metrics on manifold-valued curves.

Study reveals weak knotting in confined polymers, not dominated by any single knot type.

Holomorphic curves in $SL_2(\mathbb{C})$ embed open surfaces.

Proves Goh conditions for singular curves with specific properties.

Study uses knotoids to analyze open protein chains, revealing new topological regions.

Survey on positively curved homogeneous Finsler spaces.

Spherical quadrilaterals classified based on geometric properties.

New quantum invariants for planar knotoids improve knot classification.

The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular plane curves. Moreover, using braid monodromy, these can in turn be reformulated …

Smooth convergence shown for curve diffusion flows.

This paper constructs a continuous decomposition of the Sierpiński curve into acyclic continua one of which is an arc. This decomposition is then used to construct another continuous decomposition of the Sierpiński curve. The resulting decomposition space is homeomorphic to the continuum obtained from taking the Sierpi…

We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a fi…

Characterizes rotational solitons for curve shortening flow on revolution surfaces.

Researchers examine various causal structures for spacetimes with continuous metrics.

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves i…

Algorithm converts curves on ribbon surfaces to contact surgery diagrams.

Solves curve migration problem with elastic flows.

The study explores holomorphic Legendrian curves and superminimal surfaces in complex projective and sphere spaces.

A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…

We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.

The paper develops theory for holomorphic null curves in SL2(C).

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…

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…

It is known by A. Loi and R. Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L' to L such that the union of the links …

We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to overtwistedness, w…

Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $γ$ be a simple closed curve on $M$. Let $m γ$ denote the curve formed by tracing $γ$ $m$ times. We prove that if $m γ$ is contractible through curves of length less than $L$, then $γ$ is contractible through curves of length less than $L$. In …

A Riemannian manifold is called almost positively curved if the set of points for which all $2$-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: $Sp(3)/Sp(1)^2$, and two circle quotients of $Sp(3)/Sp(1)^2$. We also show the quasi-positively cu…

In this paper we prove a universal inequality describing the asymptotic behavior of support points for planar continuous curves. As corollaries we get an analogous result for tangent points of differentiable planar curves and some (partially known) assertions on the asymptotic of the mean value points for various class…

A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coheren…

Metrics on shape space are used to describe deformations that take one shape to another, and to determine a distance between them. We study a family of metrics on the space of curves, that includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics …