arXiv research

Shortest geodesic on curved spheres is no longer than 3 times the diameter.

The paper finds bounds on shortest dense curves on surfaces.

The paper finds shortest geodesic bounds on orbifolds with diameter limits.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

Study bounds the length of shortest periodic geodesics on certain curved spaces.

We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.

Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.

Sharp bounds found on shortest geodesic on punctured spheres.

In this paper, we establish upper bounds on the length of the shortest conjugator between pairs of infinite order elements in a wide class of groups. We obtain a general result which applies to all hierarchically hyperbolic groups, a class which includes mapping class groups, right-angled Artin groups, Burger--Mozes-ty…

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

Upper bound established for the length of shortest closed geodesics in hyperbolic link complements.

New bounds on shortest geodesic loops on a sphere.

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the sho…

Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat 1-unit-side equilateral triangles glued along their boundaries is a local extremum for the length of the shortest closed geodesic among the Riemannian spheres with conical singularities of fixed area. We give an alternative proof of this theore…

Shortest non-simple closed geodesics on hyperbolic surfaces found.

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.

Algorithms compute length spectra of torus graphs efficiently.

A curve around a sphere must be at least 4π long.

Study on shortest arcs on hyperbolic surfaces with boundary.

In this paper, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric|\leq 3$, volume $vol(M)>v>0$, and diameter $diam(M)<D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$. The proofs of our result are …

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

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

Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.

Study shortest geodesics on flat cone spheres with conical singularities.

In this paper we prove new upper bounds for the length of a shortest closed geodesic, denoted $l(M)$, on a complete, non-compact Riemannian surface $M$ of finite area $A$. We will show that $l(M) \leq 4\sqrt{2A}$ on a manifold with one end, thus improving the prior estimate of C. B. Croke, who first established that $l…

Generative Flow Networks solve shortest path problems in graphs.

We give a lower bound for the widths of the collars of certain short partial pants decomposition of the surface. Then we apply this to obtain upper bounds of the renormalized volume of certain Schottky manifolds in terms of the hyperbolic length of compressible curves.

For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…

The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.

Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.

Paper finds shortest geodesic paths on hyperbolic surfaces.

The study of random surfaces reveals asymptotic lengths of separating geodesics.

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

Improved bounds on geodesic lengths in Riemannian surfaces.

Length spectra for Riemannian metrics are well studied, while sub-Riemannian length spectra have been largely unexplored. Here we give the length spectrum for a canonical sub-Riemannian structure attached to any compact Lie group by restricting its Killing form to the sum of the root spaces. Surprisingly, the shortest …

In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed geodesic on the surface. This will follow from a more general bound applying to…

Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold M of dimension n. According to a celebrated theorem by J.P. Serre there exist infinitely many geodesics between x and y. The length of the shortest of these geodesics is obviously less than the diameter of M. But what can be said about…

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics…

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex F…

Study shows shorter words for group elements in surface groups and RAAGs.

The study finds a bound on the shortest geodesics in hyperbolic 3-manifolds.

New algorithm reduces regret in stochastic shortest path problems.

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…