Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.
problem Behavior of geodesics on cones over arbitrary Riemannian manifolds.
method Show existence of first integrals uniquely determining geodesics.
result Geodesic flow on cones is superintegrable and Liouville--Arnold integrable for non-radial trajectories.
Study connects contact structures to cone geodesics and contactomorphisms.
problem Understanding contact structures on cone geodesics.
method Review and generalize cone geodesics to contact manifolds, establish correspondence with contactomorphisms.
result Established correspondence between contactomorphisms and cone structures.
The paper describes geodesics on a Kähler cone of the Heisenberg group.
problem Understanding geodesics on the Kähler cone of the Heisenberg group.
method Analyzing the Heisenberg group to describe geodesics.
result It is not a complete manifold.
This paper studies geodesics and uniqueness of cscK cone metrics.
problem Uniqueness of constant scalar curvature Kahler cone metrics.
method Introduction of weighted function spaces, construction of cone geodesics, detailed asymptotic analysis of cscK cone metrics, linear theory for Lichnerowicz operator.
result The cscK cone metric is unique up to automorphisms.
Lightlike hypersurfaces in cone structures minimize time.
problem Finding time-minimizing paths in cone structures.
method Defining lightlike hypersurfaces and proving their foliation by cone geodesics.
result Lightlike hypersurfaces in globally hyperbolic spacetimes are time-minimizing.
New method classifies geodesics on cones.
problem Classifying geodesics on cones in 3D space.
method Using necessary and sufficient conditions for rectifying curves and their traces in spheres.
result Established conditions for geodesics on cones.
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
Study shortest geodesics on flat cone spheres with conical singularities.
problem Understanding the distribution of shortest geodesics on flat cone spheres.
method Proved a recurrent relation on the distribution of the length of shortest geodesics with respect to Thurston's volume form.
result Proved a recurrent relation on the distribution of the length of shortest geodesics.
Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length …
Study coning totally geodesic boundaries of hyperbolic manifolds.
problem Understanding metrics on coned-off spaces of hyperbolic manifolds.
method Analyzing the geometric and group-theoretic properties of coned-off spaces.
result Explicit conditions for negatively curved metrics and locally convex subsets.
Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown, if the cone structure is regarded as a control system, then, the space of abnormal geodesics of the cone structure is naturally identified with the origi…
Sharp inequalities for curved surfaces and cones.
problem Optimizing areas in nonpositively curved spaces.
method Proving inequalities for disks and triangles in cones.
result Minimal area properties for specific shapes in cones.
The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.
problem Existence of constant scalar curvature Kähler metrics with cone singularities.
method Equivalence to properness of log K-energy and geodesic stability. result Extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjecture to cscK cone metrics.
Classifies surfaces with zero mean curvature in a light cone.
problem Classifying surfaces with zero mean curvature in a light cone.
method Examined geodesics and screw motions, used Weierstrass representations.
result Complete classification of ruled zero mean curvature surfaces.
Researchers express spectral determinants on hyperbolic cones.
problem Express spectral determinants on 2D hyperbolic cones.
method Explicitly expressed spectral determinants in terms of cone angle and geodesic radius.
result Results in recent paper by Freixas i Montplet and von Pippich were incorrect.
Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…
We embed directed acyclic graphs using hyperbolic spaces and geodesic cones.
problem Learning graph representations that preserve hierarchical structure.
method Use hyperbolic spaces and geodesic cones to define embeddings of directed acyclic graphs.
result Our method significantly outperforms existing approaches in graph representation learning.
Study of cone structures and Finsler metrics linking geometric and physical aspects.
problem Defining and characterizing cone structures and Finsler metrics.
method Systematic study of cone structures and Lorentz-Finsler metrics, introducing cone triples and cone geodesics.
result Explicit descriptions of all Finsler spacetimes, including stationary and static ones.
Study transverse measures on infinite type hyperbolic surfaces.
problem Characterize the cone of transverse measures on infinite type hyperbolic surfaces.
method Use inverse limits and geodesic laminations to describe and construct cones of transverse measures.
result Explicit descriptions and bases of cones of transverse measures exist for many laminations.
The paper explores transformations between power law problems and geodesics on cones.
problem Solving power law problems and understanding their geometric properties.
method Geometric transformations and cone metrics.
result Derivation of Maclaurin duality and Jacobi-Maupertuis metric reformulation.
We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an applicat…
In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics $\mathcal H_\b$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space def…
We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic γ in such a torus is said to be homologically maximizing if one (hence every) lift of γ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics…
Study on geodesics proving index and intersection bounds, with examples of multiplicity.
problem Understanding the index and intersections of min-max geodesics on surfaces.
method Proof of tangent cone structure, construction of metrics with multiplicity.
result Upper bounds on index and intersections, examples of multiplicity.
A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly …
Proves compatibility of light cones and projective structures.
problem Clarifying different concepts of compatibility between conformal and projective structures.
method Analyzes compatibility criteria introduced by Ehlers-Pirani-Schild and Trautman-Scholz.
result Proves that the compatibility criterion introduced by Ehlers-Pirani-Schild is correct.
Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S^2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be …
An extra large metric is a spherical cone metric with all cone angles greater than 2 pi and every closed geodesic longer than 2pi. We show that every two-dimensional extra large metric can be triangulated with vertices at cone points only. The argument implies the same result for Euclidean and hyperbolic cone metrics, …
Paper proves Hölder continuity of tangent cones in RCD(K,N) spaces.
problem Understanding the geometry of metric measure spaces with curvature-dimension condition.
method Developed a second order interpolation formula for distance function.
result Tangent cones from rescalings are Hölder continuous along geodesics.
The study quantifies geodesic divergence on Riemannian planes with bounded geometry.
problem Understanding geodesic divergence on Riemannian planes with specific geometric constraints.
method Recalling quasi-redirection and using it to quantify geodesic divergence, compactifying Riemannian planes into D2 or S2. result Necessary and sufficient conditions for the quasi-redirecting compactification being S2 are derived in terms of asymptotic cones. Study on higher-dimensional quasigeodesics in metric spaces.
problem Understanding asymptotic structure of Morse quasiflats.
method Proving asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity.
result Morse quasiflats exhibit Euclidean volume growth rigidity.
New method connects compression bodies through cone manifolds.
problem Understanding how compression bodies can be transformed.
method Using cone manifold holonomy groups and standard CAT(0) space techniques. result Realized all edges in compression body graph through paths.
For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone …
We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.
In a seminal paper published in 1968, J. Simons proved that, for n≤5, the Euclidean (minimal) cone CM, built on a closed, oriented, minimal and non totally geodesic hypersurface Mn of Sn+1 is unstable. In this paper, we extend Simons' analysis to {\em warped} (minimal) cones built over a close…
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
The paper proves Schauder estimates on cone products and characterizes harmonic functions.
problem Proving Schauder estimates for metric products of cones.
method Characterizing harmonic functions and using local approximations to measure Hölder continuity.
result Interior Schauder estimates for the Laplacian on cone products are proven.
We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles ≤π), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotien…
The study proves sub-Riemannian manifolds cannot satisfy CD conditions unless they are Riemannian.
problem Characterizing sub-Riemannian manifolds that satisfy CD conditions. method Analysis of tangent cones and geodesics, construction of new RCD structures. result Sub-Riemannian manifolds are never CD(K,N) unless they are Riemannian. Study circular foliations and shear-radius coordinates on hyperbolic cone surfaces.
problem Characterize Teichmüller spaces of hyperbolic cone surfaces.
method Construct circular foliations and shear-radius coordinates on Teichmüller spaces of hyperbolic cone surfaces.
result Shear-radius coordinates provide global coordinates on Teichmüller spaces and converge to specific metrics.
Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively-curved cone surfaces. Steps are tak…
We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic c…
Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.
problem Existence and non-existence of specific geodesic nets on flat spheres.
method The theorem of Gauss-Bonnet is applied to demonstrate results.
result Existence and non-existence of geodesic nets on regular doubled polygons.
Study of generalized cones in Lorentzian geometry with causality and curvature analysis.
problem Understanding causality and curvature in Lorentzian warped products.
method Analyzing generalized cones as Lorentzian length spaces with explicit descriptions and metric curvature bounds.
result Prove singularity theorems for non-positive lower timelike curvature bounds.
We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. We construct in particular an analytic path of complete, finite-volume cone four-manifolds Mt that interpolates between two hyperbo…
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.
The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.
problem Understanding the structure and regularity of Einstein 5-manifolds.
method Analysis of tangent cones, gap theorems for 4-dimensional orbifolds, and careful metric analysis.
result Noncollapsed limits of Einstein 5-manifolds have unique and isolated tangent cones.
Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.
problem Serrin's overdetermined problem in convex cones of Riemannian manifolds.
method Rigidity results, soap bubble theorem, Heintze-Karcher inequality, drift Laplacian analysis.
result Characterization of intersections of geodesic balls with cones in Riemannian manifolds.