arXiv research

Classifies ancient convex curves in convex domains.

Flow deforms locally convex curves to curves of constant k-order width.

Compact, non-convex curve flows are created.

Convex curves evolve into circles over time.

Curve shortening in metric-affine plane shrinks convex curves to points.

The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.

Flow deforms locally convex curves into target curves.

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

In this paper we provide a characterization for a class of convex curves on the 3-sphere. More precisely, using a theorem that decomposes a locally convex curve on the 3-sphere as a pair of curves on the 2-sphere, one of which is locally convex and the other is an immersion, we are capable of completely characterize a …

The paper studies curve shortening flows on non-convex surfaces.

Two flows for convex curves converge to circles smoothly.

Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…

Classifies ancient flows in a disc with boundary.

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…

Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.

New proofs given for space curves with totally positive torsion.

For a convex curve in an even-dimensional affine space we introduce a series of convex domains (called Young hulls), describe their structure and give a formulas fo the volume of the biggest of these domains. This paper is an attempt to generalize the classical isoperimetric inequality for the volume of the convex hull…

The study explores isoptic curves of cycloids and their applications.

A simple closed curve $γ$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(γ)-2δ(γ)=3$ for anti-convex curves, where $i(γ)$ is the number of independent (true…

We prove the existence of embedded closed constant curvature curves on convex surfaces.

The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.

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.

Curve Shortening Flow preserves circularity for convex projections.

New Harnack inequality for curve shortening flow without convexity.

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…

New curve flow preserves area and converges to a circle.

We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 v…

The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.

A (positive) locally convex curve in the 2-sphere is a curve with positive geodesic curvature (i.e., which always turns left). In the 3-sphere, it is a curve with positive torsion. In this work we discussed the topology of spaces of such curves with prescribed initial and final jets. The case of the 2-sphere is underst…

Let S be a complete surface of constant curvature K = + 1 or -1, i.e. the sphere S^2 or the Lobachevskij plane L^2, and D a bounded convex subset of S. If S = S^2, assume also diameter (D) < pi/2. It is proved that the length of any steepest descent curve of a quasi-convex function in D is less than or equal to the per…

Gradient flow expands curves to round shapes.

The paper generalizes convex and star-shaped concepts to symplectic spaces and studies variational problems.

If $Γ$ is the range of a Jordan curve that bounds a convex set in $\mathbb{R}^2,$ then $\frac{1}{2}(Γ+Γ)=\mathsf{co}(Γ),$ where $+$ is the Minkowski sum and $\mathsf{co}$ is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in $\mathbb{R}^3$ with range $Γ$ such that $\frac{1}{2}(…

Paper extends Schur's theorem to spherical curves via monotonicity.

Curves with constant torsion can be deformed arbitrarily.

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{…

The paper establishes inequalities for convex curves and applies them to lattice point estimates.

Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that $C^2$ closed convex curves possess at least two tripod configurations. Later, Kao and Wang established the existence of tripod configurations for $C^2$ closed locally c…

We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call "homotopic curve shortening" (HCS), starts with a closed curve (which might self-intersect) in the presence of a set $P\subset \mathbb R^2$ of point obstacles, and evolves in discrete…

New inequalities for convex curves with multiple geometric factors.

We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.

We show that the torsion of any simple closed curve $Γ$ in Euclidean 3-space changes sign at least $4$ times provided that it is star-shaped and locally convex with respect to a point $o$ in the interior of its convex hull. The latter condition means that through each point $p$ of $Γ$ there passes a plane $H$, not cont…

We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper bound is attained when the curve intersects every plane in at most four points, a …

Suppose $τ$ is a train track on a surface $S$. Let $C(τ)$ be the set of isotopy classes of simple closed curves carried by $τ$. Masur and Minsky [2004] prove $C(τ)$ is quasi-convex inside the curve complex $C(S)$. We prove the complement, $C(S) - C(τ)$, is quasi-convex.

Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.

Study shows how a curve shortens to a half-circle under specific flow.

A curve of minimum length to enclose a unit sphere in 3D is at least 4π.