arXiv research

Quasiregular curves are Hölder continuous and have higher integrability.

Study on CR curves in 3-sphere, focusing on critical curves integration and existence.

Mathematical formulas for elliptic curve integrals solve anomaly equations.

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related w…

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

New method for curve comparison using iterated integrals and moving frames.

Using convex integration we give a constructive proof of the well-known fact that every continuous curve in a contact $3$-manifold can be approximated by a Legendrian curve.

Constructs coordinate systems from spectral curve sheaves.

Formalizes integral curves on Banach manifolds in Lean.

Gradient flow preserves speed for integral Menger curvature curves.

Integrable flows on null curves in anti-de Sitter 3-space studied.

Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some non-trivial densities on a curve…

Formula derived for Gromov-Witten invariants of smooth curves.

Study null curves and their motion in 3D flat space-time, leading to integrable hierarchies.

Curves with constant torsion can be deformed arbitrarily.

Derives path-integrals for superstrings on curved backgrounds using string geometry theory.

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize them as invariant curves under the integrable flow on plane curves which is gov…

Study extends geodesic curvature formula to higher dimensions.

The aim of this paper is to determine criteria of being integral curve for the geodesic spray of the natural lift curves of the spherical indicatrices of the involutes of a given spacelike curve with a timelike binormal in Minkowski 3-space. Furthermore, some interesting results about the spacelike evolute curve with t…

Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…

We construct integrable hierarchies of flows for curves in centroaffine ${\mathbb R}^3$ through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …

We introduce a novel class of rotation invariants of two dimensional curves based on iterated integrals. The invariants we present are in some sense complete and we describe an algorithm to calculate them, giving explicit computations up to order six. We present an application to online (stroke-trajectory based) charac…

We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is a type of moving incidence relation. The characterisation is used to provide a very general theory and construction of quantities that are necessarily conse…

We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension $2$, $3$ or $4$. We study the linearly-full stationary curves in a four-…

We consider evolution equations for curves in the 3-dimensional sphere $S^3$ that are invariant under the group $SU(2,1)$ of pseudoconformal transformations, which preserves the standard contact structure on the sphere. In particular, we investigate how invariant evolutions of Legendrian and transverse curves induce we…

In this study, we introduce a new approach to curve pairs by using integral curves. We consider the direction curve and donor curve to study curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct partner curves of a unit speed curve and giv…

The paper defines singular evolutoids and uses them to derive an integral equality.

The Hodge star mean curvature flow on a 3-dimensional Riemannian or pseudo-Riemannian manifold is a natural nonlinear dispersive curve flow in geometric analysis. A curve flow is integrable if the local differential invariants of a solution to the curve flow evolve according to a soliton equation. In this paper, we sho…

New integral expression quantizes Arnold strangeness.

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation the…

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-b…

The study classifies singularities in curved 3D shapes.

In this paper, we introduce a new approach to non-lightlike curve pairs by using integral curves in Minkowski 3-space. We consider direction curve and donor curve to study non-lightlike curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct…

Analyzes vector fields in polytope decompositions, proving curve finiteness.

In this note, we study the integral of the 1-form $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over certain plane curves defined by A-polynomials of knots. It is quite surprising that a Chern-Simons type invariant of 3-manifolds, which can be geometrically computed, may be used to get the exact values of those integrals. Th…

Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.

Conditions for simple closed curves in surface covers.

Constructs universal local deformations for curves and differential forms.

Study extends convexity in curved spaces using fractional integrals.

Derives path integrals for perturbative strings on various backgrounds.

The optimality of the integral inequality $\int\limits_γ\sqrt{k_1^2+k_2^2+k_3^2}ds>2π$ for closed curves with non-vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive curvatures in $\mathbb R^4$ satisfies the inequality $\int\limits_γ\sqrt{k_1^2+k_2^2+k_3^2}ds…

In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their Gauss maps. Further examples are found in conformal geometry, e.g. the curved flats…

We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the correspond…

We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in e…

Motion of curves and surfaces in $\R^3$ lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that…

New method linearizes Darboux transformations of discrete curves.

Contact Lie systems analyze integral curves of Hamiltonian vector fields.