arXiv research

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

Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.

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

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves' lengths do we really need to know? It is a result of Duchin--Leininger--Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We genera…

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

New method simplifies ideal curve flow with length constraint.

Flow on curves in inversive geometry converges to loxodromics.

Study on stable translation lengths of surface homeomorphisms and their approximations.

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted th…

Convex curves evolve into circles over time.

Sharp proof of sub-Riemannian length-minimizing curves being at least $C^2$

Extends curve functions to geodesic currents with a simple criterion.

The study counts curves on a once-punctured torus with self-intersections.

Study approximate marked length spectrum rigidity in non-positively curved groups.

Characterizes curves with short representatives on hyperbolic surfaces.

The study examines translation lengths of pseudo-Anosov maps on curve graphs.

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

ReLU networks don't exponentially distort curve lengths as previously thought.

Study finds minimum lengths of curves on a one-holed torus.

Three methods solve spatial rational curves with rational arc length.

Generalizes Toponogov theorem to Alexandrov spaces.

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed tran…

The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any…

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 …

Study minima of geodesic lengths for specific curves on surfaces.

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…

We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by L + epsilon. If the manifold is orientable, then for an…

We prove that for every analytic curve in the complex plane, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Euc…

Study on abnormal curves in sub-Riemannian manifolds, proving length-minimizing properties.

In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in the plane. If $K$ is locally-connected, connected and compact, then the level set…

New proof shows rationality of scl for non-filling curves.

The paper studies stability of discrete planar curves using variational methods.

We prove that the length difference between a closed periodic curve and its parallel curve at a sufficiently small distance is proportional to the rotation index. As an application, the rotation index of a curve could be estimated by means of Cauchy-Crofton formula.

We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a low…

The paper analyzes discrete approximations to minimize curve length in Euclidean space.

We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giv…

The paper studies how curves evolve under area constraints and converges to a critical point.

In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.

New findings on translation lengths in Teichmüller and curve graphs for pseudo-Anosovs.

Here a new notion of fractional length of a smooth curve, which depends on a parameter $σ$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length u…

Study on elastic curves with variable stiffness, derived from bending energy.

Study curves evolving on hypersurfaces with free boundaries, preserving length.

In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group of Euclidean space (translations, rotations, and reflections). In equiaffine ge…

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.

Method calculates systolic length of modular curves.

We prove that the Garside length a braid is equal to a winding-number type invariant of the curve diagram of the braid.

Let $Σ$ be a surface of negative Euler characteristic and $S$ a generating set for $π_1(Σ,p)$ consisting of simple loops that are pairwise disjoint (except at $p$). We show that the word length with respect to $S$ of an element of $π_1(Σ,p)$ is given by its intersection number with a well-chosen collection of curves an…