arXiv research

Geometric interpretation of 3-manifold invariants using immersed curves.

We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an applicati…

New homomorphism proven using immersed curves on disks.

Framework for isometric immersions of planar regions from framed curves.

Study disproves conjecture about metric completion of curve spaces.

We prove short-time existence of φ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.

Space curves with convex projections evolve smoothly until shrinking to a point.

In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formu…

Study proves uniqueness of corrugated negatively curved immersions in differential geometry.

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ …

A new method converts knot Floer homology to immersed curves.

We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i) biminimal curves in a Riemannian manifold, with particular care to the case of …

Study how $J^+$ invariants change under bifurcations of curves.

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…

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…

The space of all immersed closed curves of rotation degree 0 in the plane modulo reparametrizations has the same homotopy groups as the circle times the 2-sphere.

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…

Investigates Darboux rectifying curves on smooth surfaces.

In this paper, we introduce the notion of developments of curves with respect to symmetric tensors and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. Our method provides a geometric construction of such an isometric immersion.

Study character varieties of tangles to map immersed curves in the pillowcase.

Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.

Curve Shortening Flow preserves circularity for convex projections.

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also sho…

The paper proves existence of minimal homotopies for immersed planar curves.

Holomorphic curves in moduli spaces are quasi-isometrically immersed.

Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.

The paper tackles deeply slice knots via immersed curves.

A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and …

Mannheim curves are defined for immersed curves in 3-dimensional sphere S^3 . The definition is given by considering the geodesics of S^3. First, two special geodesics, called principal normal geodesic and binormal geodesic, of S^3 are defined by using Frenet vectors of a curve immersed in S^3. Later, the curve alpha i…

Curves with constant curvature are flexible and can be deformed.

Smooth curves with specific curvature can be closely approximated.

A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this artic…

Study $d$-invariants of L-space double branched covers of arborescent links.

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theor…

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserma…

We prove that every immersed $C^2$-curve $γ$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_γ$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_γ$. The approximating curves satisfy a $C^1$-dense $h$-principle. As an application we obtain the existence of $C^2$-knots of arbitrary p…

The study proves stable minimal immersions in positively curved manifolds are totally geodesic.

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 $n\geq 2$ are metrically complete on the space $\mathcal I^n(S^1,\mathbb R^d)$ of Sobolev immersions of the same regularity and that any two curves i…

Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

Suppose $δ$ is a generic immersed closed curve in the boundary of a 3-manifold M and $δ$ is null-homotopic in M. Then $δ$ can be displaced by a height function in a collar of the boundary so that the resulting simple closed curve in the collar bounds a disk in M.

New method computes knot Floer homology for satellite knots.

Given a plane curve $γ: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(γ)$ of inflections which curves $\mbox{diff}(γ)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of $\mathbb R^2$. We show that if $γ$ is an immersed curve with $D(γ)$ double points and no othe…

A divide is a relative generic immersion of a finite union of copies of the unit interval in the unit disk. A divide defines a classical link in the 3- sphere, which is a fibered link if the image of the immersion is connected. We prove in this paper, that the Lefschetz number of the monodromy is 0. This result was kno…

Proper superminimal surfaces in hyperbolic 4-space can be approximated by conformal immersions.

Study proves h-principles for curves in bracket-generating distributions.

Study on immersions with flat normal bundle in curved spaces.

We apply contact homology to obtain new results in the problem of distinguishing immersed plane curves without dangerous self-tangencies.