Geodesics with bounded angles have zero Hausdorff dimension.
problem Understanding the geometric properties of geodesics with bounded angles.
method Analyzing the Hausdorff dimension of geodesics with specific angle constraints.
result The set of geodesics with bounded self-intersection angles has a Hausdorff dimension of zero.
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 2 log ( 5 + 2 6 ) 2\log(5+2\sqrt6) 2 log ( 5 + 2 6 ) , and this bound is sharp. We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…
Estimates barycenter in geodesic spaces with finite sample bounds.
problem Estimating the barycenter of a distribution in geodesic spaces.
method Finite sample error bounds, Hoeffding- and Bernstein-type concentration inequalities, efficient algorithms.
result Statistical guarantees for efficient barycenter computation.
New method to bound Laplacian eigenvalues of geodesic balls.
problem Computing upper bounds for the first eigenvalue of Laplacian on geodesic balls.
method Transforming metric tensor into rotationally symmetric form preserving geodesic sphere areas.
result Upper bound for Laplacian eigenvalues is sharp and computable using geodesic sphere areas.
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.
Improved lower bound for geodesics on manifolds.
problem Finding a lower bound for the number of minimal geodesics on Riemannian manifolds.
method Refined Bangert's method using the stable norm unit ball on the first homology.
result Quadratic lower bound for the number of minimal geodesics.
Study bounds changes in hyperbolic 3-manifold structures after drilling short geodesics.
problem Bounding changes in complex projective structures after drilling short geodesics.
method Analyzes L 2 L^2 L 2 -bounds on changes in conformally compact hyperbolic 3-manifolds. result Change is bounded by a universal constant times the square root of the length of the drilled geodesics.
The study finds a bound on the shortest geodesics in hyperbolic 3-manifolds.
problem Finding bounds on the shortest geodesics in hyperbolic 3-manifolds.
method Establishing an upper bound for the length of the n t h n^{th} n t h shortest closed geodesic in terms of the volume of the manifold. result An upper bound for the length of the n t h n^{th} n t h shortest closed geodesic in terms of the volume of the manifold. Sharp lower bound found for geodesic ball eigenvalues.
problem Finding the minimum eigenvalue for geodesic balls.
method Applied Li-Schoen's uniform Poincare inequality for non-negative Ricci curvature manifolds.
result Sharp lower bound of the first Dirichlet eigenvalue for geodesic balls.
Upper bound on geodesic ball volume in Riemannian manifolds.
problem Bounding geodesic ball volume in Riemannian manifolds.
method Techniques to provide an upper bound on geodesic ball volume.
result Upper bound on geodesic ball volume in Euclidean space.
Bounds on geodesic distances on Stiefel manifold derived from new metrics.
problem Improving geodesic computation algorithms and understanding Stiefel manifold.
method New geometric insights and Lipschitz constants for geodesic distances.
result Explicit bounds on geodesic distances and conditions for attaining bounds.
Study shows volumes of knot complements are bounded by linear functions of geodesic periods.
problem Volume calculation of knot complements associated with geodesics on modular surfaces.
method Analyzes geodesics on modular surfaces, their associated knots, and their complements' volumes.
result Volumes of knot complements are bounded linearly by the period of geodesic continued fractions.
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.
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 k e x t t h k^ ext{th} k e x t t h -shortest geodesic is at most 8 k d 8kd 8 k d . Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.
problem Finding a sharp lower bound for the first Dirichlet eigenvalue of geodesic balls.
method Quantitative explicit inequality linking the width of geodesic balls to the spectral gap.
result A quantitative inequality relating the width of geodesic balls to the spectral gap between the first Dirichlet eigenvalue and its lower bound.
Superdense flows on surfaces imply bounded geodesics, and vice versa.
problem Understanding the relationship between superdense flows and bounded geodesics on translation surfaces.
method Analyzing Teichmüller geodesics and their associated flows on translation surfaces.
result A linear flow on a translation surface is superdense if and only if the associated Teichmüller geodesic is bounded.
Uniform bound on geodesic images for surfaces using bicorn curves.
problem Bounding geodesic images on closed surfaces.
method Utilizing bicorn curves and properties of 1-slim triangles.
result Uniform bound of 44 for non-annular subsurfaces, 3 for specific cases.
The study bounds the number of closed geodesics in a specific orbit closure of surfaces.
problem Counting closed geodesics in a specific orbit closure of surfaces.
method Analyzes triangulations and Teichmüller geodesics to bound the number of closed geodesics.
result Obtains exponential bounds on the number of closed geodesics of length at most R.
In this paper we establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the n n n -sphere. We also bound the length of certain minimizing geodesic nets.
Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.
problem Regularity of distance maps on geodesically complete spaces with curvature bound above.
method Define and prove regularity of distance maps as Hurewicz fibrations.
result Sphere theorem for geodesically complete CAT(1) spaces.
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.
Paper bounds total geodesic curvature using boundary data in hyperbolic gravity.
problem Bounding total geodesic curvature in a hyperbolic setting.
method Derives an upper bound for total geodesic curvature in terms of boundary data.
result Upper bound for total geodesic curvature expressed solely in terms of boundary data.
We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for …
A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear in the geometric length of the geodesic.
For the pants graph, there is little known about the behaviour of geodesics, as opposed to quasigeodesics. Brock-Masur-Minsky showed that geodesics or geodesic segments connecting endpoints satisfying a bounded combinatorics condition, such as the stable/unstable laminations of a pseudo-Anosov, all have bounded combina…
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.
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.
Proves conjecture about geodesic foliations in Riemannian planes.
problem Geodesic foliations with bounded distance in non-flat Riemannian planes.
method Analyzes total curvature and visibility properties to prove conjecture.
result Proves conjecture in two specific cases.
Lower bound found for volumes of modular link complements.
problem Finding a lower bound for the volumes of modular link complements.
method Analyzing the volumes of link complements associated with geodesics in the modular surface.
result First linear lower volume bound in terms of exponents of code words.
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.
Geometric bounds for low Steklov eigenvalues on hyperbolic surfaces with boundaries.
problem Finding lower bounds for low Steklov eigenvalues of hyperbolic surfaces with geodesic boundaries.
method Analysis of eigenfunction behavior on an adapted thick-thin decomposition for hyperbolic surfaces with geodesic boundaries.
result Sharp geometric lower bounds for low Steklov eigenvalues that depend on the shortest multi-geodesic disconnecting the surfaces.
Study on extremal subsets in geodesically complete spaces with curvature constraints.
problem Characterizing extremal subsets in GCBA spaces.
method Introduced and analyzed extremal subsets in GCBA spaces, proving their properties.
result Set of topological singularities forms an extremal subset under additional assumptions.
Study geodesic curvature in 2D Alexandrov spaces, generalizing results from spaces with curvature above.
problem Geodesic curvature in Alexandrov spaces with curvature below.
method Comparison and rigidity theorems for geodesic curvatures.
result Generalized known results for geodesic curvature in spaces with curvature above.
We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly hyperbolic, thus a universal bound can be given for the diameter of the geodesic image. We also generalize the theorem for pro…
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.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only C 0 C^{0} C 0 dependent on the metric coefficients.
The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.
problem Characterizing geodesics on hyperbolic surfaces.
method One-parameter Allen-Cahn min-max constructions.
result Every geodesic occurs with multiplicity one and provides uniform sharp lower bounds.
Extends Polydisk Theorem to Cartan-Hartogs domains.
problem Characterizing geodesics in Cartan-Hartogs domains.
method Applying the Polydisk Theorem to Cartan-Hartogs domains.
result Cartan-Hartogs domains inherit geodesic submanifolds from bounded symmetric domains.
We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we p…
Bounding geodesic length variation for surface projective structures.
problem Understanding how geodesic lengths change under projective structure variations.
method Bounding the derivative of complex length in terms of the Schwarzian norm.
result Application to cone-manifold deformations of hyperbolic 3-manifolds.
Characterizes visibility and geodesic loops in complex domains.
problem Visibility and geodesic loops in complex domains.
method Using quasi-geodesic frames to characterize visibility and geodesic loops.
result Characterizes visibility and existence of geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains.
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 k k self-intersections improved from 512 to 128. Study geodesics in 3-torus, determining complements' topology.
problem Understanding the topology of geodesic complements in 3-torus.
method Analyzes the orbit of direction vectors under P S L 3 ( Z ) PSL_3(\mathbb{Z}) P S L 3 ( Z ) action and uses Farey graph distances. result Determines homeomorphism type of geodesic complements in 3-torus.
For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geomet…
Characterizes geodesics on spheres with Morse index bounds and inequalities.
problem Understanding geodesics on spheres using Morse theory.
method Morse-theoretic characterization and strong Morse inequalities.
result Existence of geodesics with specific Morse indices on spheres.
New upper bound for geodesic complexity derived from cut locus decompositions.
problem Understanding geodesic complexity in Riemannian manifolds.
method Study of decompositions of cut loci and their tangent fibers.
result Established a new upper bound for geodesic complexity.