arXiv research

New harmonic maps to hyperbolic plane via Bäcklund transformation.

A 6-regular triangulation for hyperbolic plane created.

New conditions found for hyperbolic bicycle tracks.

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.

The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

Study inverse curve shortening flow on hyperbolic plane, classifying solitons.

The paper defines catenary curves in spheres and hyperbolic planes.

Study on closed $p$-elastic curves in hyperbolic and de Sitter planes.

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.

Study compares hyperbolic and quasihyperbolic metrics in plane domains.

Characterizes hyperbolic links with stable maps to the plane.

Classifies tilings of hyperbolic plane by regular polygons.

Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated hyperbolic plane. We also provide a model of a cylinder in a hyperbolic space. This appr…

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.

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras $\mathfrak{f}_4$ and $\mathfrak{f}^*_4$, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we cla…

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.

We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.

The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.

The hyperbolic structure of equilateral pentagons is mapped to a tiling of the hyperbolic plane.

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…

Geometric structures over algebras describe geodesics and spaces.

A discrete method approximates hyperbolic curvature flow in the plane.

The paper classifies discrete complex hyperbolic triangle groups.

Study Steklov eigenvalues on hyperbolic triangle-tiling graphs.

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 $-I/Δ^2$ 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…

New translations defined; curve shortening flow solved in hyperbolic plane.

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 closed trajectories in hyperbolic plane with specific curvature constraints.

Harmonic maps from complex plane to hyperbolic space constructed using heat flow.

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξφT=TR_ξφ$, 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.

Examines discrete curvature's relation to smooth curvature in 3 spaces.

Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical expo…

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ 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 $\mathbb{P}^2$.

Study proves unique compactification of hyperbolic space.

We show a generic finiteness result for least area planes in 3-dimensional hyperbolic space. Moreover, we prove that the space of minimal immersions of disk into hyperbolic space is a submanifold of a product bundle over a space of immersions of circle into sphere at infinity. The bundle projection map when restricted …

We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.

We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Ham…

Study quasisymmetric maps on hyperbolic plane boundaries.

The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action on R^3 by polyno…

Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homoge…