Proves Ptolemaean Inequality and Theorem in complex hyperbolic plane.
problem None explicitly stated, but related to complex hyperbolic geometry.
method Proof of Ptolemaean Inequality and Theorem in complex hyperbolic plane with Cygan metric.
result Proves Ptolemaean Inequality and Ptolemaeus' Theorem in the closure of complex hyperbolic plane.
Researchers found parallel mean curvature tori in complex projective and hyperbolic planes.
problem Finding tori with parallel mean curvature vectors.
method Explicit determination of tori in complex projective and hyperbolic planes.
result Explicit determination of tori with parallel mean curvature vectors in both complex projective and hyperbolic planes.
Characterizes hypersurfaces in complex projective and hyperbolic planes.
problem Geometric characterization of hypersurfaces.
method Geometric characterization and partial classifications.
result Characterizes certain hypersurfaces of cohomogeneity one.
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.
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.
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.
Study classifies hypersurfaces in complex hyperbolic spaces with specific operator conditions.
problem Classifying real hypersurfaces with commuting structure Jacobi operators.
method Used simultaneous diagonalization for commuting symmetric operators to classify hypersurfaces.
result Complete classification of real hypersurfaces with commuting condition.
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. 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.
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).
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 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.
Study classifies and explores isoparametric submanifolds in complex space forms.
problem Characterizing isoparametric submanifolds in complex space forms.
method Analyzes submanifolds in complex hyperbolic planes and two-dimensional complex space forms.
result Classified Terng-isoparametric submanifolds in two-dimensional complex space forms.
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.
Global conformal parameters found for complex and hyperbolic curves.
problem Finding global conformal parameters for analytic curves.
method Analytic curves in complex and hyperbolic planes, and their generalizations.
result Spherical and hyperbolic arc-lengths are global conformal parameters for analytic curves.
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.
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.
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…
This paper computes Seshadri constants for fake projective planes using the Toledo invariant.
problem Computing Seshadri constants for fake projective planes.
method Using the Toledo invariant and $\C$-Fuchsian curves in complex hyperbolic spaces.
result Explicit computation of Seshadri constants for all ample line bundles on fake projective planes.
Compactify complex hyperbolic almost Hermitian manifolds.
problem Understanding the geometric structure of complex hyperbolic almost Hermitian manifolds.
method Analyzing the asymptotic curvature and boundary conditions of the manifold.
result The interior of a compact almost complex manifold can represent the original manifold.
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.
It is of interest to characterize algebraically the dynamical types of isometries of the complex and quaternionic hyperbolic planes. In the complex case, such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of the isometries of the two-dimensional quate…
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.
The paper studies real hyperbolic complements in complex hyperbolic plane.
problem Analyzing real hyperbolic complements in complex hyperbolic plane.
method Warping functions and curvature tensor computations.
result Warping functions yield complete finite volume metrics with bounded negative curvature.
We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of…
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hami…
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H…
Few years ago we developed jointly with I.Dynnikov new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially deep results were obtained for the Euclidean plane with equilateral triangle lattice. In the present work we develop a DCA theory for …
Study geometric properties of a complex hyperbolic group action.
problem Geometric properties of a specific modular group action.
method Explicit description of subgroups and conjugacy classes.
result Explicit description of a torsion-free subgroup of index 336.
The paper constructs harmonic maps from punctured surfaces to the hyperbolic plane.
problem Harmonic maps from punctured surfaces to hyperbolic planes.
method Constructing polynomial growth harmonic maps from punctured Riemann surfaces to regular polygons in hyperbolic plane.
result Established uniqueness of harmonic maps within a class of maps differing by exponentially decaying variations.
In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,m/S(U2⋅Um), m≥2 from the equation of Gauss. Next we derive a new formula for the Ricci tensor of M in SU2,m/S(U2⋅Um). Finally we giv…
Maps complex plane polynomials to light-like polygons in Einstein Universe.
problem Mapping between complex plane polynomials and light-like polygons.
method Constructs geometric homeomorphism between moduli spaces.
result Found minimal Lagrangian maps between ideal polygons.
In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.
We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and co…
The paper studies Lagrangian surfaces in a specific Riemannian product space.
problem Characterizing and classifying Lagrangian surfaces in a particular geometric space.
method Analyzes various types of Lagrangian surfaces and their properties.
result Classification of different types of Lagrangian surfaces.
In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…
Study complex hyperbolic structures on a disc orbibundle with 5 cone points.
problem Constructing complex hyperbolic structures on a disc orbibundle with vanishing Euler number.
method Analyzing involutions in PU(2,1) and using bending-connectedness. result A 4-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle.
New manifolds with many hyperbolic planes, not homogeneous.
problem Constructing non-homogeneous manifolds with many hyperbolic planes.
method Constructing manifolds with geodesics in hyperbolic planes.
result Non-homogeneous manifolds with many hyperbolic planes exist.
Study on discrete properties of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of type [m1, m2, 0].
method Analysis of isometries generated by complex reflections in ultra-parallel geodesics.
result Proves discreteness and non-discreteness results for these groups.
Complex Hyperbolic Triangle Groups of Type [m,m,0;3,3,2]math.GT Study on discrete complex hyperbolic triangle groups of specific type.
problem Discreteness of complex hyperbolic triangle groups of type [m,m,0;3,3,2]. method Analysis of groups generated by complex reflections with specific orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
Abstract: Study of geometric structures on surfaces using various tools.
problem Understanding geometric structures on surfaces.
method Use of volume, contact, symplectic, complex, and almost complex structures; local rigidity results; higher-dimensional analogues; constructions with Riemann surfaces; definitions using surjective homomorphisms; models of hyperbolic plane and 3-space; conformal structures.
result Introduction of new models and constructions for hyperbolic plane and 3-space.
Study complex reflections in 3D hyperbolic geometry, finding new representations.
problem Deforming groups in 3D complex hyperbolic geometry.
method Representations of abstract groups in PU(3,1), using Ford domains as guides.
result First nontrivial example of a discrete and faithful representation of a subgroup in PU(3,1).
The paper finds optimal inequalities for hypersurface curvatures in complex Grassmannians.
problem Optimizing inequalities for hypersurface curvatures in complex Grassmannians.
method Analyzing real hypersurfaces in complex Grassmannians to derive inequalities involving scalar curvature and Casorati curvatures.
result Conditions for equality in inequalities involving normalized scalar curvature and Casorati curvatures.
We classify all of real hypersurfaces M with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians SU2,m/S(U2⋅Um), m≥2. Then it becomes a tube over a totally geodesic SU2,m−1/S(U2⋅Um−1) in SU2,m/S(U2⋅Um) or a horosphere whose center at infinity is …
We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to…
New representations of 3-manifold groups into complex hyperbolic space found.
problem Finding representations of 3-manifold groups into complex hyperbolic spaces.
method Using Lefschetz fibrations and orbifold fundamental groups of branched coverings of the projective plane.
result Infinitely many non-conjugate representations discovered.