arXiv research

The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.

We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the $ε_0$-relative part of Teichmuller spaces of surfaces with boundary.

Study self-intersections of arcs on a pair of pants, proving natural number spectrum.

There are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal Teichmüller space, the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller sp…

We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined m…

We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comp…

Given a compact boundaryless Riemannian manifold $Y$ on which a compact Lie group $G$ acts, there is always a metric on $Y$ such that the action is by isometries. Assuming $Y$ is equipped with such a metric, recall that the $G$-invariant Laplacian is the restriction of the ordinary Laplacian to the space of functions w…

The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.

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 present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical con…

Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length [Cristea\&Steinsky 2009…

Three methods solve spatial rational curves with rational arc length.

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…

This paper constructs PH spline curves with prescribed arc lengths.

Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.

Let α(s) be an arc on a connected oriented surface S in E3, parameterized by arc length s, with torsion τ and length l. The total square torsion F of α is defined by T=\int_{0}^{l}τ^{2}ds\ $. . The arc α is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the v…

Let $α$ be an arc on a connected oriented surface $S$ in Minkowski 3-space, parameterized by arc length $s$, with torsion $τ$ and length $l$. The total square torsion $H$ of $α$ is defined by $$. The arc is called a relaxed elastic line of second kind if it is an extremal for the variational prob…

The method for approximation of planar curve by circular arcs with length preservation, proposed by I.Kh. Sabitov and A.V. Slovesnov, is analyzed. We extend the applicability of the method, and consider some corollaries, not related to the approximation problem. Inequalities for the length of a convex spiral arc with p…

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers $δ>0$ which identify the distinct $δ$ covers of the space. We investigat…

Paper estimates optimal ROC curve arc length and AUC, improving classification performance.

We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in Mio-Srivastava-Joshi on the space of arc-length parameterized loops. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of …

Projected random forests improve circular data prediction with adaptive arc length and finite-sample coverage.

Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.

Study on length spectrum of random hyperbolic 3-manifolds.

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a confor…

New method proves length spectrum rigidity in various geometric settings.

New theorem shows metrics of certain groups are close if their lengths are identical.

We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a Poisson algebra homomorphism from C(S) to the space of smooth functions on the decorated Teichmuller space endowed with …

The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada [17] and many other authors this result is far from being true for the unmarked…

Anosov surfaces with same length spectrum are isometric.

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stabilit…

Study shows surfaces with similar length spectra are smoothly deformable.

Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the length-spectrum Teichm…

Curve diffusion flow straightens curves with endpoints on intersecting lines.

Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.

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

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…

We show that the Teichmüller space of a surface without boundary and with punctures, equipped with Thurston's metric is the limit (in an appropriate sense) of Teichmüller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the tran…

Study on shortest arcs on hyperbolic surfaces with boundary.

New proof shows Fuchsian groups have irrational length spectra.

In this article, we prove that every arithmetic locally symmetric orbifold of classical type without Euclidean or compact factors has arbitrarily long arithmetic progressions in its primitive length spectrum. Moreover, we show the stronger property that every primitive length occurs in arbitrarily long arithmetic progr…

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This prope…

We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.

We will develop simple relations between the arc-lengths of a pair of geodesics that share common end-points. The two geodesics differ only by the requirement that one is constrained to lie in a subspace of the parent manifold. We will present two applications of our results. In the first example we explore the converg…

Study of lengths of cycles in large genus random maps converging to Poisson process.

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic…