Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.
problem Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
method Applied Perron's method and Thurston's algorithm to prove existence and convergence.
result Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
Circle's metric is at least π/4 away from any simply connected geodesic space.
problem Comparing simply connected geodesic spaces to the circle.
method Using Gromov-Hausdorff distance and topological properties.
result The Gromov-Hausdorff distance between circle and any simply connected geodesic space is at least π/4.
Study geodesics in constrained curve spaces, including elastic curves and concentric circles.
problem Geodesics in constrained curve spaces with Sobolev metrics.
method Intrinsic and constructive approaches.
result Construct geodesics in elastic curve and concentric circle spaces.
Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
Closed geodesics densely cover a circle in dilation surfaces.
problem Density of closed geodesics in dilation surfaces.
method Study of Teichmüller flow and Delaunay triangulation.
result Directions of closed geodesics are dense in the circle.
Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.
problem Whether a prescribed total geodesic curvature can be realized by a degenerated circle packing.
method Introduced combinatorial Ricci flow to find the desired degenerated circle packed surface, analogous to Chow-Luo and Takatsu methods.
result Fully characterized sufficient and necessary conditions for the existence of degenerated circle packings and showed their uniqueness.
Paper resolves spherical curvature flow problem.
problem Existence of ideal circle patterns in spherical background geometry.
method Introduces a combinatorial geodesic curvature flow in spherical background geometry.
result Characterizes sufficient and necessary conditions for flow convergence.
New method finds ideal circle patterns on spheres.
problem Finding ideal circle patterns on spheres with prescribed curvatures.
method Combinatorial Calabi flow in spherical geometry.
result Existence and convergence of the flow for ideal circle patterns.
Persistent homology reveals geometric features of metric spaces, especially geodesic circles.
problem Detecting geometric features in metric spaces using persistent homology.
method Analyzing algebraic elements (footprints) in persistent homology of metric spaces and subspace.
result Higher-dimensional persistent homology captures lower-dimensional geometric features.
Self dual symmetric R-spaces have special curves, called circles, introduced by Burstall, Donaldson, Pedit and Pinkall in 2011, whose definition does not involve the choice of any Riemannian metric. We characterize the elements of the big transformation group G of a self dual symmetric R-space M as those diffeomorphism…
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
This note treats the notion of Lagrange derivative for the third order mechanics in the context of covariant Riemannian geometry. The variational differential equation for geodesic circles in two dimensions is obtained. The influence of the curvature tensor on the Lagrange derivative leads to the emergence of the notio…
We classify all Kahler metrics in an open subset of C2 whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on CP2 restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the E…
We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an S1-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain light…
Study local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.
problem Understanding local diffeomorphisms of conformal circles.
method Variations of conformal circles and pseudo-Riemannian manifolds.
result Local diffeomorphisms of conformal circles are conformal local diffeomorphisms.
The paper solves circle packings on surfaces with boundaries.
problem Circle packing on surfaces with boundaries and finite genus.
method Using Thurston's algorithm and discrete Schwarz-Pick lemma.
result A unique solution to the boundary value problem exists.
We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). …
We construct all Finsler metrics on the two-sphere for which geodesics are circles and show that any (reversible) path geometry on a two-dimensional manifold is locally the system of geodesics of a Finsler metric.
In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol3. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in tot…
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Theorem shows generic metrics yield non-degenerate geodesic nets.
problem Characterizing geodesic nets on generic metrics.
method Proving all connected embedded nets are non-degenerate for Baire-generic metrics.
result All stationary geodesic nets are non-degenerate for generic metrics.
The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.
problem Understanding the behavior of geodesics and measures on fibered hyperbolic 3-manifolds.
method Properties of geodesics and measures on the circle and sphere are analyzed to prove singularity.
result Natural measures on the circle become singular with respect to measures on the sphere.
Paper constructs hyperbolic metrics using circle packings and curvature parameters.
problem Creating polyhedral metrics for surfaces of various topologies.
method Using circle packings and curvature parameters, the paper constructs hyperbolic polyhedral metrics.
result Unified approach to producing polyhedral metrics for surfaces of broader topological types.
Compactifies geodesic flows on hyperbolic surfaces, revealing attractive circles at infinity.
problem Geodesic flows on non-compact hyperbolic surfaces without cusps.
method Constructs a geometrical compactification using one-dimensional distributions tangent to stable and unstable horocycles.
result Existence of attractive circles at infinity in the compactified flow.
We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is …
The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be r…
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
Asymptotics for equidistribution of circles on hyperbolic surfaces.
problem Equidistribution of circles on hyperbolic surfaces.
method Spectral method and statistical limit theorems.
result Precise asymptotics for the rate of equidistribution of circles.
R-circles in general three dimensional CR manifolds (of contact type) are the analogues to traces of Lagrangian totally geodesic planes on the sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain legendrian curves on the manifold. We prove that a diffeomorphism betwee…
Slim curves on 3-sphere help spherical CR uniformizations.
problem Understanding curves on 3-sphere for CR uniformizations.
method Defining slimness, analyzing foliations, and applying to quasi-Fuchsian groups.
result Slim curves lead to spherical CR uniformizations of certain 3-manifolds.
In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit…
We show that on a closed Riemannian manifold with fundamental group isomorphic to Z, other than the circle, every isometry that is homotopic to the identity possesses infinitely many invariant geodesics. This completes a recent result of the second author.
Following the lines of a celebrated result by R. Bott (Comm. Pure Appl. Math. 9, 1956) we study the Morse index of the iterated of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, w…
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
We consider a closed orientable Riemannian 3-manifold (M,g) and a vector field X with unit norm whose integral curves are geodesics of g. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of g. We study when this 2-plane bundle remains i…
The paper examines flows that preserve area and length in hyperbolic geometry.
problem Preserving area and length in hyperbolic geometry.
method Inverse curvature flows for convex curves in hyperbolic plane.
result The flows converge to geodesic circles under certain conditions.
A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we apply Lie derivatives and the Cartan Y-connection to study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on t…
We study geodesics on the modular surface, comparing WP and hyperbolic metrics.
problem Comparing geodesics on the modular surface under different metrics.
method Lift WP geodesics to the universal cover, analyze geometric properties, and compare deviations.
result WP and hyperbolic geodesics fellow-travel in the thick part of the universal cover.
The paper solves curvature problems on infinite hyperbolic surfaces.
problem Prescribed curvature flow on hyperbolic surfaces with infinite topological type.
method Introduced a prescribed curvature flow adapted to infinite cellular decompositions.
result Established well-posedness of the flow and proved convergence under certain conditions.
Maps and measures on surfaces link best Lipschitz and least gradient functions.
problem Analyzing maps between surfaces and their geometric properties.
method Duality between best Lipschitz and least gradient maps, geodesic laminations, and transverse measures.
result The infinity harmonic map defines a geodesic lamination and the least gradient map defines a transverse measure.
The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.
problem Adapting results for Reeb flows and Hamiltonian flows with closed orbits.
method Adapting results from Geodesic circle foliations to Reeb and Hamiltonian flows.
result All orbits on connected contact manifolds with closed orbits have identical periods.
Wave maps from circle to manifold controllable if homotopy classes match.
problem Global controllability of wave maps from circle to Riemannian manifolds.
method Characterization of controllability via homotopy classes, uniform-time global controllability between steady states, quantitative exponential stability.
result Global controllability is equivalent to homotopy class of data.
The two main topics of this text are as follows: Firstly, three modifications of the theorem of Beltrami will be presented for diffeomorphisms between Riemannian manifolds and a space form which preserve the geodesic circles, the geodesic hyperspheres, or the minimal surfaces, respectively. Secondly, it is defined what…
Study infinite combinatorial Ricci flow on spherical surfaces.
problem Investigate infinite combinatorial Ricci flow with spherical background.
method Establish existence and convergence of solution for infinite cellular decompositions.
result Existence and convergence of solution for infinite combinatorial Ricci flow in spherical geometry.