New harmonic maps to hyperbolic plane via Bäcklund transformation.
problem Constructing new harmonic maps to the hyperbolic plane.
method Using Bäcklund transformation to connect solutions of sinh-Gordon and sine-Gordon equations.
result Construction of new harmonic maps.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
New conditions found for hyperbolic bicycle tracks.
problem Conditions for hyperbolic bicycle tracks.
method Hyperbolic development interpretation of bicycling monodromy.
result New necessary and sufficient conditions for hyperbolic bicycle tracks.
The study finds Lie algebra formulae and classifies polar actions on a hyperbolic plane.
problem Finding Lie algebra formulae and classifying actions on hyperbolic planes.
method Using octonions and triality, explicit Lie brackets were found for Lie algebras of isometry groups.
result Explicit formulae for Lie brackets of f4 and f4∗ Lie algebras. Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.
problem Understanding isolation properties of geodesic planes in hyperbolic 3-manifolds.
method Quantitative estimates of geodesic planes in frame bundles, using tight areas and densities.
result Polynomial estimates of isolation properties with degree given by modified critical exponents.
We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.
Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.
problem Existence of harmonic maps from product of hyperbolic planes to hyperbolic space.
method Existence result for asymptotic Dirichlet problem.
result Existence of harmonic maps with given boundary data.
The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.
Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.
problem Characterizing limit sets of knots in complex hyperbolic geometry.
method Analyzing embeddings of knots as limit sets of discrete subgroups of PU(2, 1).
result Knots are either chains or R-circles as limit sets.
Study inverse curve shortening flow on hyperbolic plane, classifying solitons.
problem Understanding the behavior of curves in hyperbolic geometry under a specific flow.
method Classifying solitons with respect to vector fields and studying their properties.
result Parabolic solitons are graphs on the y-axis, conformal solitons on the x-axis.
The paper defines catenary curves in spheres and hyperbolic planes.
problem Defining catenary curves in non-Euclidean geometries.
method Characterizations of catenary curves in terms of curvature and angle with geodesics.
result Characterizations and extensions of catenary curves in hyperbolic geometry.
Study on closed p-elastic curves in hyperbolic and de Sitter planes.
problem Existence of closed p-elastic curves with nonconstant curvature. method Analysis of p-elastic curves in hyperbolic and de Sitter planes. result Existence of closed p-elastic curves in hyperbolic plane for p>1; in de Sitter plane for p<0. In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.
Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.
problem Outer billiards in complex hyperbolic plane.
method Symplectic form and geodesic foliation analysis.
result Outer billiard map is a diffeomorphism and symplectomorphism.
New metric on torus Teichmüller space is isometric to hyperbolic plane.
problem Realizing hyperbolic plane as moduli space of marked Riemann surfaces.
method Defined metric on Teichmüller space of torus, used extremal Lipschitz maps.
result Teichmüller space of torus with new metric is isometric to hyperbolic plane.
Classifies tilings of Euclidean and hyperbolic planes using topological methods.
problem Classifying crystallographic tilings of Euclidean and hyperbolic planes.
method Topological approaches, including enumeration and classification of tilings as decorations of orbifolds.
result Classification up to equivariant equivalence achieved.
Study compares hyperbolic and quasihyperbolic metrics in plane domains.
problem Comparing hyperbolic and quasihyperbolic metrics in plane domains.
method Analyzes metric spaces and boundaries of hyperbolic domains, proving equivalence and constructing counterexamples.
result Hyperbolic and quasihyperbolic metric spaces are quasiisometrically equivalent for finitely connected hyperbolic domains, but not in general.
Characterizes hyperbolic links with stable maps to the plane.
problem Understanding hyperbolic links through stable maps.
method Characterization of hyperbolic links via stable maps to the plane.
result Complete characterization of hyperbolic links with specific stable maps.
Classifies tilings of hyperbolic plane by regular polygons.
problem Decidability of tiling patterns in hyperbolic plane.
method Finite set of local and inductive combinatorial constraints.
result First known weakly aperiodic protosets of regular polygons in hyperbolic plane.
Study of curves in hyperbolic plane with variable curvature.
problem Finding curves with prescribed almost constant curvature in hyperbolic plane.
method Analyzing closed and embedded curves with geodesic curvature.
result Existence of curves with specified curvature variations.
Crochet models show hyperbolic geometry properties.
problem Visualizing hyperbolic geometry concepts.
method Triangulated hyperbolic plane models.
result Demonstrates various hyperbolic geometry properties.
This is an expository article about groups generated by two isometries of the complex hyperbolic plane.
We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.
In this paper, we prove two results. First, there is a family of sequences of embedded quarters of the hyperbolic plane such that any sequence converges to a limit which is an end of the hyperbolic plane. Second, there is no algorithm which would allow us to check whether two given ends are equal or not.
Study hyperbolic geometry to find Fibonacci numbers.
problem Investigate exponential growth and isoperimetric inequality in hyperbolic geometry.
method Combinatorial approximation of hyperbolic plane, calculations.
result Surprising link to Fibonacci numbers.
The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.
problem Constructing Funk-Finsler structures in hyperbolic models.
method Using Finsler isometries and explicit computations, the Funk-Finsler structure is constructed in various hyperbolic models.
result The Funk-Finsler structure in the Klein unit disc is a Randers metric.
The hyperbolic structure of equilateral pentagons is mapped to a tiling of the hyperbolic plane.
problem Mapping the realization space of equilateral pentagons to a hyperbolic plane.
method Combining combinatorial correspondence, Riemann mapping theorem, and normalization procedure.
result A full conformal parameterization of the space of equilateral pentagons.
We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce v…
We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.
Geometric structures over algebras describe geodesics and spaces.
problem Understanding geodesics in hyperbolic and Euclidean spaces over non-standard algebras.
method Study geometric structures from Hermitian forms on real algebras (dual numbers, split-complex, split-quaternions).
result Presented a projective model for the hyperbolic bidisc.
A discrete method approximates hyperbolic curvature flow in the plane.
problem Modeling wave phenomena in solid-liquid interfaces.
method Semidiscrete finite difference method for hyperbolic curvature flow.
result Error bounds for natural discrete norms are proven.
The paper classifies discrete complex hyperbolic triangle groups.
problem Classifying discrete complex hyperbolic triangle groups.
method Analyzing complex hyperbolic spaces and isometries.
result Classifies discrete complex hyperbolic (n,∞,∞)-triangle groups for n=3,4,5. Geodesic planes in 3-manifolds are either closed or dense.
problem Characterize geodesic planes in geometrically finite acylindrical 3-manifolds.
method Analyzes geodesic planes in the interior of convex cores of 3-manifolds.
result Geodesic planes are either closed or dense, with countably many closed ones.
Study Steklov eigenvalues on hyperbolic triangle-tiling graphs.
problem Analyzing Steklov eigenvalues on specific hyperbolic graph structures.
method Introduced a graph roughly isometric to hyperbolic plane, used discretization to transfer bounds.
result Steklov eigenvalues tend to zero proportionally to the inverse of the domain size.
Shorter sides in geodesic triangles in hyperbolic plane.
problem Properties of geodesic triangles in hyperbolic surfaces.
method Analyzing lifts of a closed geodesic in hyperbolic 2-space.
result Sides of triangles formed by geodesics are shorter than the geodesic itself.
New translations defined; curve shortening flow solved in hyperbolic plane.
problem Solving curve shortening flow in hyperbolic geometry.
method Introduced new translations, solved equations, analyzed ancient solutions.
result Explicit solutions and area estimates for ancient solutions.
We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a three-body problem in the Euclidean plane. The potential is −I/Δ2 where I is the triangle's moment of inertia and Δ its area. The reduction method uses the Jacobi-Maupertuis metric, following the author's earlier p…
In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize "ideal cones" (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.
Study of special elliptic isometries and their lengths in complex hyperbolic plane.
problem Classifying lengths of special elliptic isometries in complex hyperbolic plane.
method Classification and description of relative SU(2,1)-character varieties.
result Fully classified lengths of special elliptic isometries (2, 3, 4).
Study of closed trajectories in hyperbolic plane with specific curvature constraints.
problem Critical trajectories in hyperbolic plane for a specific energy function.
method Classification of critical trajectories based on momentum causal character, proof of existence of closed trajectories.
result Existence of countably many closed trajectories with time-like momentum.
Harmonic maps from complex plane to hyperbolic space constructed using heat flow.
problem Constructing harmonic maps from complex plane to hyperbolic space.
method Heat flow method to construct harmonic maps.
result Harmonic maps are unique once the principal part of their Hopf differential is prescribed.
In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field T, that is, RξφT=TRξφ, where T=A or T=S for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators…
Study finds optimal loops in hyperbolic space with Finsler structure.
problem Optimal loops in Finsler hyperbolic plane.
method Left-invariant Finsler structure, convex trigonometry functions.
result Optimal isoperimetric loops found in terms of trigonometry functions.
Study lifts plane mappings to Heisenberg group.
problem Contact quasiconformal mappings in hyperbolic Heisenberg group.
method Lifting Theorem for symplectic mappings.
result Symplectic mappings lifted to Heisenberg group.
Examines discrete curvature's relation to smooth curvature in 3 spaces.
problem Understanding how discrete curvature relates to smooth curvature in different spaces.
method Using specific triangular tilings of 3 types of spaces to examine curvatures.
result Discrete curvature can sense the smooth curvature of ambient space forms.
Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.
problem Classify complete totally geodesic subsets of complex hyperbolic plane.
method Two new proofs: one algebraic and one geometric.
result Only complex geodesics and real planes are non-trivial complete totally geodesic subsets.
In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane E2 were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…
First example of a hyperbolic 4-orbifold underlying P2.
problem Finding closed hyperbolic 4-orbifolds with symplectic underlying spaces.
method Realized P2 as the underlying space of a closed hyperbolic 4-orbifold. result First example of a closed hyperbolic 4-orbifold with symplectic underlying space.