Upper bounds on shortest filling geodesics on hyperbolic surfaces.
problem Finding the shortest geodesic that covers a surface.
method Quantitative density of closed geodesics on hyperbolic surfaces.
result Upper bounds on the length of the shortest closed geodesic.
The paper finds bounds on shortest dense curves on surfaces.
problem Finding shortest dense curves on surfaces.
method Quantitative density of closed geodesics and orthogeodesics.
result Upper bounds on shortest dense curves.
This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.
problem Classifying and determining the length of the shortest filling pairs on a specific type of surface.
method Classifying and determining the length of the shortest filling pairs on a specific type of surface.
result The paper classifies and determines the length of the shortest minimal filling pairs on a genus two surface.
The article finds conditions for separating filling pairs on surfaces and constructs a Morse function.
problem Conditions for separating filling pairs on surfaces.
method Combinatorial study of mapping class group action and construction of a Morse function.
result Cardinality of global minima of the Morse function equals the number of orbits of filling pairs.
The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.
problem Finding the asymptotic behavior of shortest closed multi-geodesics on hyperbolic surfaces.
method Analyzing the length of shortest filling closed multi-geodesics using hyperbolic geometry and asymptotic analysis.
result The length of a shortest filling closed multi-geodesic is uniformly comparable to a specific formula involving the genus and lengths of closed geodesics.
Let Sg denote the closed orientable surface of genus g. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill Sg and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric g…
Paper finds shortest geodesic paths on hyperbolic surfaces.
problem Finding the shortest geodesic paths on hyperbolic surfaces.
method Analyzes genus g hyperbolic surfaces to find minimal length geodesics.
result Minimal geodesic length is realized by a specific polygon.
Origami edge-paths connect coherent curves on surfaces.
problem Understanding coherent curves on surfaces.
method Origami structure and edge-paths.
result Existence of origami edge-paths connecting coherent curves.
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
Study compares hyperbolic and extremal lengths for shortest curves.
problem Comparing hyperbolic and extremal lengths for shortest curves.
method Lower bounds for widths of collars and upper bounds for renormalized volume of Schottky manifolds.
result Upper bounds of renormalized volume in terms of hyperbolic length of compressible curves.
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.
The study explores different definitions of geodesics in sub-Riemannian geometry.
problem Understanding geodesics in sub-Riemannian geometry and their equivalence.
method Review of three variational definitions of geodesics and three definitions of straightest curves.
result Shortest geodesics coincide with straightest geodesics in some sub-Riemannian manifolds.
Algorithms find second and third shortest non-trivial closed walks on surfaces.
problem Finding non-trivial closed walks on surfaces efficiently.
method Algorithms based on careful analysis of shortest curves and configurations.
result Second shortest walk found in O(n2logn) time, third in O(n3) time. A curve around a sphere must be at least 4π long.
problem Finding the shortest closed curve that encloses a sphere.
method Analyzing curves in Euclidean 3-space and comparing their lengths.
result The shortest curve is composed of 4 semicircles arranged like a baseball seam.
Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.
problem Quantifying the complexity of non-simple closed geodesics on hyperbolic surfaces.
method Analyzing the geometry of shortest figure eight curves and constructing geodesic representatives.
result Explicit upper bounds for the length of shortest geodesics with k self-intersections improved from 512 to 128. Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.
problem Finding shortest non-simple closed geodesics disjoint from orbifold points.
method Fundamental domains and hyperbolic trigonometry.
result Identified and classified all figure eight geodesics on triangle group orbifolds.
A behavior of extreme networks under deformations of their boundary sets is investigated. It is shown that analyticity of a deformation of boundary set guarantees preservation of the networks types for minimal spanning trees, minimal fillings and so-called stable shortest trees in the Euclidean space.
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 generalizes Lebesgue curves to new space-filling and fractal sets.
problem Generating space-filling curves from planar substitutions.
method Generalized Lebesgue's construction to new curves and fractal sets.
result Some substitutions create relatively dense fractal-like sets.
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 8π(n−1). A curve of minimum length to enclose a unit sphere in 3D is at least 4π.
problem Finding the shortest closed curve that encloses a unit sphere within its convex hull.
method Analyzing the geometric properties and using convex hull concepts.
result The minimum length of such a curve is 4π in 3D, with equality in a specific case.
Small sets of systoles fill hyperbolic surfaces of large genus.
problem Constructing hyperbolic surfaces with minimal systole sets.
method Theory of Coxeter groups combined with number theory.
result Cardinality of systole sets is in o(g/ ln g) for large genus.
New proof shows how to fill a square with Fibonacci curve.
problem How to fill a square with a space-filling curve.
method Used Cartesian product of Fibonacci substitution with itself.
result Different proof of space-filling curve construction.
A closed hyperbolic surface of genus g≥2 can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. The…
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.
New proof shows rationality of scl for non-filling curves.
problem Understanding stable commutator length in non-filling curves.
method New proof using extremal surfaces for scl.
result Rationality of stable commutator length for non-filling curves.
Logarithmic growth in random walk projections and shortest curves in mapping tori.
problem Understanding the growth of random walk projections and shortest curves in mapping tori.
method Analyzing random walks and their projections, applying to hyperbolic groups and Out(F_n).
result The shortest geodesic in a mapping torus has length on the order of 1/ log^2(n).
The study of symplectic fillings for rational cuspidal curves.
problem Understanding symplectic fillings of contact manifolds associated with rational cuspidal curves.
method Exploration through Stein handlebodies and rational blow-downs.
result Examples of contact manifolds that are links of normal surface singularities, and those that do not admit symplectic fillings.
Study of a beam sliding freely along two curves.
problem Finding the shortest curve with free sliding endpoints.
method Geometric formulation on smooth manifolds with boundary.
result Rigorous geometric approach to free boundary problems.
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
Paper finds minimal number of curves in surface systems.
problem Determining minimal number of curves in surface systems.
method Analyzes oriented surfaces of genus g for positive integers k and g.
result Exact minimal number of curves in filling k-systems.
New method for constructing space-filling curves for self-similar sets.
problem Constructing space-filling curves for self-similar sets.
method Skeleton concept and neighbor graph analysis.
result Connected self-similar sets satisfying the finite type condition always possess skeletons.
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.
New proof confirms surfaces can be divided into polygons.
problem Estimating the dimension of Thurston spine.
method Existence of an infinite set A, proving codimension is o(g/ log g).
result Proves recent conjecture about Thurston spine dimension.
Study fills a surface with odd, non-3 curves.
problem Determining filling pairs for surfaces with odd, non-3 punctures.
method Constructed minimally intersecting pairs of curves.
result Completed the classification of filling pairs for all surfaces.
Algorithm decides if geodesic curves are filling on surfaces.
problem Determining if geodesic curves are filling on surfaces.
method Efficient algorithm using Dehn-Thurston coordinates and combinatorial length bounds.
result Explicit bound for combinatorial length in terms of hyperbolic length.
In this paper we deepen the analysis of certain classes M_{g,k} of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of M_{g,k} is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusp…
Sharp bounds found on shortest geodesic on punctured spheres.
problem Finding the shortest closed geodesic on punctured spheres.
method Sharp curvature-free upper bounds expressed in terms of area, extremal metrics described.
result Optimal bounds for spheres with up to four ends, extended to larger numbers of punctures.
Study minima of geodesic lengths for specific curves on surfaces.
problem Finding the shortest geodesic paths on surfaces.
method Using curves related to dessins d'enfants and Grothendieck-Belyi surfaces.
result Minima of geodesic lengths are achieved on Riemann surfaces defined over number fields.
New bounds on shortest geodesic loops on a sphere.
problem Finding shortest geodesic loops on a sphere.
method Analyzing geodesic loops starting and ending at a fixed point on a sphere.
result At any point on a sphere, there are at least two distinct geodesic loops whose lengths are bounded by 8d and 14d.
Transformers adapted to spherical geometry using space-filling curves.
problem Generalizing transformers to geometric domains like spheres.
method Attention heads following a space-filling curve.
result Introduction of the Spiroformer on a 2-sphere.
We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.
New Stein fillings found for rational surface singularities.
problem Exploring Stein fillings of rational surface singularities.
method Using planar open books and Lefschetz fibrations, describe Stein fillings via symplectic disk arrangements.
result Many rational singularities admit Stein fillings not diffeomorphic to Milnor fibers.
Study on the minimum length of curves on once-punctured hyperbolic surfaces.
problem Finding the minimum length of filling pairs on once-punctured hyperbolic surfaces.
method Analyzing the topology and geometry of the surface to derive a lower bound for the length of filling pairs.
result A lower bound for the length of filling pairs on once-punctured hyperbolic surfaces is derived, depending only on the surface's topology.
Optimizes the first eigenvalues of Riemann surfaces for large genus.
problem Finding optimal lower bounds for first eigenvalues of Riemann surfaces.
method Analyzing shortest multi-closed curves to establish a new lower bound.
result The first eigenvalue of a Riemann surface is greater than a specific formula involving the genus and a constant.
Spirals are not shortest paths in certain sub-Riemannian geometries.
problem Nonminimality of spiral-like curves in sub-Riemannian manifolds.
method Construction of a competing curve to demonstrate non-minimality.
result Spiral-like curves are not length minimizing in sub-Riemannian manifolds.
We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of punctures. We are able to establish orders of growth for even k and show that for…