arXiv research

We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

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

New approach to extremal hyperbolic surfaces using NEC groups.

We describe in this paper a geometric construction in the projective p-adic plane that gives, together with a suitable notion of p-adic convexity, some open subsets of P 2 .Q p / naturally endowed with a "Hilbert" distance and a transitive action of PGL.2; Q p / by isometries. ese open sets are natural analogues of the…

The paper proves a Penrose inequality for 4-discs with hyperbolic ends and conic singularities.

New 2D complex hyperbolic structures found on sphere orbibundles.

Study complex hyperbolic structures on a disc orbibundle with 5 cone points.

Develops orbibundle theory for complex hyperbolic geometry.

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

New complex structure on hyperbolic disc within hyperkaehler space.

Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.

The study shows that certain complex geometries are hyperbolic and contractible but fail to be CAT(0).

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to de…

Sharp estimate on harmonic maps at conformal points in balls.

Groups satisfy linear surface isoperimetric functions.

Minimal discs count as knot invariants in hyperbolic 4-space.

We present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphas…

Plumbing of surfaces embeds in hyperbolic 4-manifolds.

Applying the techniques developed in [AGG], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [GLT], such bundles $M\to S$ should satisfy the inequality $|eM/χS|\leqslant1$, where $eM$ stands for the E…

We prove that the embedding of the quaternionic hyperbolic disc $H^1_\mathbb{H}$ into quaternionic hyperbolic $n$-space $H^n_\mathbb{H}$ is tight and thereby obtain the value of the Gromov norm of the quaternionic Kähler class.

We establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in ${\rm PSU}(1,\,1)$. As an application, we construct explicitly a new class of hyperbolic metrics …

In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the bou…

This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface $Σ$ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open …

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as an application of the minimal surface theory. This looks an interesting phenome…

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…

Study models Indian stock market using hyperbolic geometry for market stability and volatility analysis.

Study knot groups from disc patterns in 3D space.

The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…

We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on $\mathcal{U}$ that make it homeomorphic to $\mathbb{R}^2$ . By considering the actio…

Geometric structures over algebras describe geodesics and spaces.

Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence of a second extremal packing in the same surface, but constructing explicit examp…

Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a π_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has a natural compactification as a closed disc that inherits a π_1 action. The embedd…

A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A boundary-value problem for this example is formulated and is shown to possess we…

If M is a hyperbolic 3-manifold with a quasigeodesic flow then we show that π_1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is Möbius-like but not conjugate into PSL(2, R). We conjecture that the latter possibility…

We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these proper…

Linear progress observed in fibered 3-manifold embeddings.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

The paper studies the automorphism groups of specific 3-manifolds and finds upper bounds for their sizes.

In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing th…

Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.

New knots found with tough, unsliceable discs.

Hyperbolic geometry reveals financial network structure and systemic importance.

Classifies ancient flows in a disc with boundary.