New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,694 papers · 148 categories
Trend · papers per month
48 results for “geodesic length”
This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.
problem Understanding relationships between geodesic and orthogeodesic lengths on hyperbolic surfaces.
method Investigates a broad family of identities involving lengths of all closed geodesics and orthogeodesics.
result Introduces new identities that include lengths of all closed geodesics, contrasting with previous identities.
In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length , where 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…
Study geodesic paths on flat surfaces, comparing length and singularity counts.
problem Comparing geometric length and singularity counts on geodesic paths.
method Apply counting limit laws to infinite graphs and then to flat surfaces.
result Statistical comparison of geometric length and singularity counts on geodesic paths.
Improved bounds on geodesic lengths in Riemannian surfaces.
problem Finding precise lengths of geodesics in Riemannian surfaces.
method Proved curvature-free linear length bounds on geodesics.
result Length of -shortest geodesic is at most .
From curves to currentsmath.GT
Extends curve functions to geodesic currents with a simple criterion.
problem Continuous extension of curve functions to geodesic currents.
method Simple criterion based on smoothing property.
result Extends known curve functions and introduces new examples.
Lower bounds on geodesic length with few intersections on hyperbolic surfaces.
problem Finding the minimum length of geodesics with at least 2 intersections.
method Analyzing geodesics on hyperbolic surfaces with at least 2 self-intersections.
result The minimum length of such geodesics is , and this bound is sharp.
Analyzes geodesic lengths in sparse networks, deriving a distribution.
problem Understanding connectivity and robustness in networked systems.
method Analytic derivation of geodesic length distribution in sparse networks.
result Simple closed-form expression for geodesic length distribution.
Stability of Convex Spheresmath.DG
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short () geodesics on a …
We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the Krawtchouk polynomials, we give many pairs of p-isospectral flat manifolds having di…
Let be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmüller space of the surface using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the leng…
Shortest geodesic on curved spheres is no longer than 3 times the diameter.
problem Finding the shortest closed geodesic on spheres with positive curvature.
method Proved a new isoperimetric inequality for spheres with pinched curvature, used to improve the bound on the shortest geodesic.
result The shortest closed geodesic is no longer than 3 times the diameter of the sphere.
Upper bound established for the length of shortest closed geodesics in hyperbolic link complements.
problem Finding bounds on the length of shortest closed geodesics in hyperbolic link complements.
method Established an upper bound for the length of an nth shortest closed geodesic as a logarithmic function of the volume of the manifold.
result An upper bound of the length of an nth shortest closed geodesic is established as a logarithmic function of the volume of the manifold.
Study geodesics on graphs with random lengths, proving bi-infinite paths exist.
problem Existence of bi-infinite geodesic paths on graphs with random edge lengths.
method Sublinear Morse geodesics and first passage percolation analysis.
result Proves the existence of bi-infinite geodesic paths in graphs with specific properties.
On a hyperbolic Riemann surface, given two simple closed geodesics that intersect times, we address the question of a sharp lower bound on the length attained by the longest of the two geodesics. We show the existence of a surface on which there exists two simple closed geodesics of length interse…
Bounding Shortest Closed Geodesics with Diameter on compact 2-dimensional Orbifolds Homeomorphic to math.DG
The paper finds shortest geodesic bounds on orbifolds with diameter limits.
problem Bounding the shortest closed geodesic on compact orbifolds.
method Generalizing length-bounded sweepouts to orbifolds.
result Established an inequality linking shortest geodesic length to orbifold diameter.
Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric…
Geodesic currents on surfaces have comparable metrics in thick regions.
problem Comparing the geometry of geodesic currents and their minimizing metrics.
method Analyzing the space of geodesic currents on surfaces and comparing metrics on thick components.
result Geometries of geodesic currents and their minimizing metrics are comparable in thick regions.
The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.
problem Uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.
method Introduced the notion of timelike marked length spectrum and constructed length-twist coordinates.
result Uniqueness of closed timelike geodesics in their free homotopy class.
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
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.
The paper proves geodesics and conic sections are length-minimizing under specific metrics.
problem Finding shortest paths in complex geometries.
method Calibrations and conformal metrics.
result Geodesics and conic sections are length-minimizing.
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…
Study bounds the length of shortest periodic geodesics on certain curved spaces.
problem Bounding the length of shortest periodic geodesics on curved spaces.
method Analyzing the space of closed loops and their homotopy.
result The length of a shortest periodic geodesic is bounded by .
New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.
problem Identifying surfaces by their geodesic lengths.
method Analyzes metrics on simple, thick negatively curved two-dimensional P-manifolds.
result Piecewise negatively curved Riemannian metrics on simple, thick two-dimensional P-manifolds are uniquely determined by their geodesic lengths.
Minimal geodesics on hyperbolic surfaces are long.
problem Finding the shortest closed geodesics on hyperbolic surfaces.
method Analyzing the self-intersection number to estimate geodesic lengths.
result The minimal length of geodesics grows logarithmically with the self-intersection number.
Let be a Riemannian -sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on . In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed , where is the diameter of . We a…
Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
problem Understanding the relationship between intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
method Analyzing the asymptotic behavior of interaction strength I(X) as X approaches infinity in the moduli space of compact hyperbolic surfaces.
result Determined the asymptotic behavior of interaction strength I(X) in terms of the length of the shortest geodesic sys(X).
The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.
problem Uniform discreteness and linear dependence of geodesic lengths in arithmetic orbifolds.
method Analyzes Salem numbers and Lie groups to prove uniform discreteness, and uses geometric properties to show linear dependence of geodesic lengths.
result Existence of a positive constant δ(X) such that squares of lengths of closed geodesics shorter than δ must be pairwise linearly dependent over Q.
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.
Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.
problem Bounding the length of shortest closed geodesics on Riemannian manifolds with good covers.
method Generalization of previous results using diameter, volume, and cover elements to bound geodesic length.
result Length of shortest closed geodesic is bounded by a function of volume, diameter, and cover elements.
The paper finds an upper limit for the length of geodesic chords on Riemannian manifolds.
problem Finding the maximum length of geodesic chords on Riemannian manifolds.
method Establishing an upper bound for geodesic chord length using geometric bounds on the manifold.
result An upper bound for the length of geodesic chords is derived, with a specific example for 2-dimensional spheres.
Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.
problem Linking number of Legendrian knots on negatively curved surfaces.
method Analyzes Poincaré series on negatively curved surfaces.
result Explicit rational value of Poincaré series at 0 interprets linking number of Legendrian knots.
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…
Counting hyperbolic multi-geodesics with individual component lengths.
problem Counting hyperbolic multi-geodesics with specific component lengths.
method Unified geometric and topological techniques, combining Mirzakhani's results and Margulis's ideas.
result Asymptotic polynomial counts of multi-geodesics in mapping class group orbits, generalizing Wolpert's conjecture.
3D spheres can't be swept by short curves, complicating geodesic length estimates.
problem Obstructing geodesic length estimates in 3D spheres.
method Constructing specific 3D spheres with controlled diameter and volume.
result Min-max methods for geodesic lengths fail for certain 3D spheres.
Thurston introduced shear deformations (cataclysms) on geodesic laminations - deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined…
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
Research extends geodesic length function study to three holed sphere.
problem Geodesic length function on orbifolds.
method Extending previous work on punctured torus to three holed sphere and related orbifolds.
result Extension to three holed sphere and related orbifolds.
Study counts geodesics on hyperbolic 3-manifolds, proving prime theorems.
problem Counting primitive closed geodesics on compact hyperbolic 3-manifolds.
method Proves prime geodesic theorems with symmetric error terms in length and holonomy.
result Effective equidistribution of holonomy and symmetric error terms.
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.
Lower bounds on geodesic lengths for spheres with Willmore energy.
problem Finding shortest closed geodesics on spheres with Willmore energy.
method Proving a lower bound on geodesic lengths for spheres with Willmore energy below 6π.
result The energy threshold of 6π is optimal and the inequality cannot be extended to higher genus surfaces.
On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.
Study on abnormal curves in sub-Riemannian manifolds, proving length-minimizing properties.
problem Characterizing abnormal geodesics in sub-Riemannian manifolds.
method Analyzing curves that annihilate Lie brackets and proving minimization properties.
result Strictly abnormal geodesics can cease to be locally length-minimizing.
Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.
problem Properties of geodesics on flat tori and their lengths.
method Fine study of dynamical correlation function and anisotropic Sobolev spaces.
result Properties of the length distribution and singularities of its Fourier transform.
Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.
problem Properties of geodesics on expander surfaces of high genus.
method Adapting Margulis' counting strategy to low length scales.
result Almost every geodesic of certain lengths is filling or non-simple.
Improved bounds on geodesic intersections on hyperbolic surfaces.
problem Finding the shortest geodesic with a specific number of intersections.
method Proved a new formula for minimal length of geodesics with self-intersection number k.
result Improved the threshold for the existence of geodesics with self-intersection number k.