arXiv research

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

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.

New proof shows surfaces can have identical length spectra but not simple ones.

We show that certain families of iso-length spectral hyperbolic surfaces obtained via the Sunada construction are not generally simple iso-length spectral.

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…

Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$. We a…

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect $n$ times, we address the question of a sharp lower bound $L_n$ on the length attained by the longest of the two geodesics. We show the existence of a surface $S_n$ on which there exists two simple closed geodesics of length $L_n$ interse…

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…

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

A simple method reduces bias in LLM auto-evaluators by controlling output length.

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…

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Fi…

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

We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly det…

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

Extends curve functions to geodesic currents with a simple criterion.

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 …

Proves stability of convex spheres with similar geodesic lengths.

The length of shortest non-simple geodesics grows logarithmically with surface genus.

Study shows Transformers can generalize to varying task lengths.

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.

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

We describe a new approach to the study of the set of all simple geodesics on a hyperbolic punctured torus. We introduce a valuation on the first integral homology group of the torus. This valuation associates to each homology class the length of the unique simple geodesic in it. We show that this valuation extends to …

New bounds on specific torsion lengths for periodic mapping classes.

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given…

Reformulated Markov's conjecture in combinatorial terms.

Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order $o(\log\log g)$ with $g$…

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.

The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.

Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Lein…

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…

New bounds on shortest geodesic loops on a sphere.

Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.

We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.

We prove an analogue of Farb-Masur's theorem that the length-spectra metric on moduli space is "almost isometric" to a simple model $\mathcal {V}(S)$ which is induced by the cone metric over the complex of curves. As an application, we know that the Teichmüller metric and the length-spectra metric are "almost isometric…

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…

A simple text model shows word lengths follow Zipf's law.

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…

We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spe…

We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.

Study on the minimum length of curves on once-punctured hyperbolic surfaces.

We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thur…

Mirzakhani obtained the asymptotic growth, when $L\to\infty$, of the number of curves in the mapping class group orbit of some given simple curve and with length at most $L$. Years later she extended this result from simple to arbitrary curves. Here we give a short and relative low-tech argument showing how to derive t…

Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a…

It is proved that the stable commutator length of a Dehn twist in the mapping class group is positive and the tenth power of a Dehn twist about a nonseparating simple closed curve is a product of two commutators. As an application a new proof of the fact that the growth rate of a Dehn twist is linear is given.