Develops orbibundle theory for complex hyperbolic geometry.
problem Calculating invariants of complex hyperbolic disc orbibundles.
method Using diffeology, calculates Euler number and Toledo invariant.
result New tools for invariants of complex hyperbolic orbibundles.
Study minimal surfaces in complex hyperbolic space, linking entropy and volume.
problem Characterize minimal submanifolds in complex hyperbolic space.
method Analyze asymptotic regularity and introduce Colding-Minicozzi entropy and CR-volume.
result Establish a connection between Colding-Minicozzi entropy and CR-volume.
We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometr…
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic surfaces and their compactifications.
The paper builds complex hyperbolic 2-manifolds with isolated singularities.
problem Finding compact complex hyperbolic 2-manifolds with non-free actions and isolated fixed points.
method Constructs specific examples for each prime p with Z/pZ action. result General examples for p=2 related to complex hyperbolic lattices conjugacy separability. The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the modular group and, more generally, triangle groups. These are some of the simpl…
Study reveals hidden accidental parabolics in complex hyperbolic geometry.
problem Understanding hidden parabolics in complex hyperbolic geometry.
method New technique to show ideal boundary of Ford domain is an infinite-genus handlebody.
result 3-manifold at infinity of Δ4,∞,∞;∞ is the complement of chain link 814. Study of quaternionic hyperbolic space bisectors and their decompositions.
problem Understanding bisectors in quaternionic hyperbolic geometry.
method Developed theory of quaternionic bisectors, showed various decompositions, derived projection formulas.
result Introduced fan decompositions of quaternionic bisectors by totally geodesic submanifolds isometric to complex hyperbolic space.
This paper classifies Ricci solitons in complex hyperbolic spaces.
problem Understanding Ricci solitons in complex hyperbolic spaces.
method Analyzing homogeneous expanding Ricci solitons as submanifolds of complex hyperbolic spaces.
result Classification and analysis of Lie subgroups with Ricci soliton induced metric in complex hyperbolic spaces.
Study of holonomy algebras and Kahler homogeneous structures in complex hyperbolic spaces.
problem Characterizing holonomy algebras and Kahler homogeneous structures in complex hyperbolic spaces.
method Detailed investigation of holonomy algebras and their actions, using canonical connections and Tricerri-Vanhecke's classification.
result Formula for all Kahler homogeneous structures on complex hyperbolic spaces and related holonomy properties.
The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We e…
Complex hyperbolic triangle groups are discrete when certain conditions are met.
problem Proving discreteness of complex hyperbolic triangle groups.
method Analyzing representations and isometries to determine discreteness.
result Conditions for discreteness of complex hyperbolic triangle groups are identified.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In th…
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
problem Complex hyperbolic geometry
method Algebraic invariant calculus
result Denominator-cleared identities for various geometric quantities
We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.
Complex Hyperbolic Triangle Groups of Type [m,m,0;n1,n2,2]math.GT The study determines discreteness of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of specific type.
method Analysis of groups generated by complex reflections with given orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.
The paper explores symplectic geometry of Cartan-Hartogs domains.
problem Understanding the symplectic geometry of Cartan-Hartogs domains.
method Constructing a dual counterpart and computing symplectic capacity.
result A Cartan-Hartogs domain admits symplectic duality if and only if it reduces to a complex hyperbolic space.
Holonomy groups of complex hyperbolic submanifolds are always transitive.
problem Understanding the restricted normal holonomy groups of complex hyperbolic submanifolds.
method Lifting to pseudo-Riemannian space, developing new tools for degenerate submanifolds, introducing weakly polar actions.
result The restricted normal holonomy group is always transitive for complex hyperbolic submanifolds.
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
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. 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.
Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.
problem Characterizing manifolds at infinity of complex hyperbolic orbifolds.
method Spherical CR uniformization and Dehn surgery.
result Specific 3-manifolds at infinity of complex hyperbolic triangle groups are identified.
Researchers mapped the moduli space of a specific group in 3D complex hyperbolic geometry.
problem Mapping the moduli space of a discrete, faithful representation of the modular group in PU(3,1). method Constructed the entire moduli space M by parameterizing it with a square, relating it to PU(2,1) representations. result The moduli space M is divided into subspaces parameterized by a square, each corresponding to different geometries. Sharp asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
problem Understanding the asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
method Analyzing the curvature and metric properties of Kähler-Einstein metrics on complex hyperbolic cusps.
result Sharp doubly exponential rate of convergence of metrics to a limiting form.
The study proves a tube theorem for complex hyperbolic manifolds.
problem Understanding the geometry of complex hyperbolic manifolds.
method Tubular neighborhood theorem and geometric combination theorem.
result Explicit estimates and bounds for tube widths in complex hyperbolic manifolds.
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).
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.
We consider the space M of ordered quadruples of distinct points in the boundary of complex hyperbolic n-space, chn, up to its holomorphic isometry group PU(n,1). One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for M. For $n=2…
3-manifolds are CR uniformized on spheres, proving a conjecture.
problem Uniformizing 3-manifolds with cusps using CR methods.
method Spherical CR uniformization of complex hyperbolic triangle groups.
result Magic 3-manifolds and other cusped 3-manifolds are CR uniformizable.
Study complex reflections in infinite Coxeter tetrahedron moduli space.
problem Characterize representations of Coxeter group in complex hyperbolic space.
method Type-preserving representations of Coxeter group G to PU(3,1), parameterized by θ. result Discrete and faithful representations for θ∈[65π,π]. First nontrivial moduli space in complex hyperbolic space. New examples of real hypersurfaces found in complex hyperbolic quadrics.
problem Existence of specific types of real hypersurfaces in complex hyperbolic quadrics.
method Construction of a one-parameter family of homogeneous Hopf hypersurfaces.
result First known examples of real hypersurfaces with integrable maximal complex subbundle in irreducible Kahler manifolds.
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 …
Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension 2. We prove a Mertens' formula for the integer points over a quadratic imaginary num…
Classifies CR submanifolds in complex hyperbolic spaces.
problem Understanding CR submanifolds in complex hyperbolic spaces.
method Classifying orbits of a subgroup of the solvable part of the Iwasawa decomposition.
result Classification of homogeneous CR submanifolds.
Compactifies CR structures for complex hyperbolic manifolds.
problem Building compact CR structures for complex hyperbolic manifolds.
method Constructs a compactification by a strictly pseudoconvex CR structure.
result Establishes a compact CR structure for asymptotically locally complex hyperbolic manifolds.
Study describes moduli of quaternionic hyperbolic triples of points.
problem Tackles the congruence classes of triples of points in quaternionic hyperbolic space.
method Introduces invariants and defines quaternionic Goldman invariants for mixed configurations.
result Defines quaternionic analogues of Goldman invariants for mixed configurations.
We describe and study the loci equidistant from finitely many points in the so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic 2-ball B. In particular, we show that the bisectors (= the loci equidistant from 2 points) containing the (smooth real algebraic) curve equidistant from gi…
These are lectures on discrete groups of isometries of complex hyperbolic spaces, aimed to discuss interactions between the function theory on complex hyperbolic manifolds and the theory of discrete groups.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into PU(n,1) and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sulliva…
Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.
problem Classifying ruled real hypersurfaces with constant norm.
method Analyzing nonflat complex space forms, proving existence and uniqueness.
result Existence of a unique inhomogeneous example in complex hyperbolic space.
New interpretation of complex hyperbolic form as Weil-Petersson form.
problem Understanding complex hyperbolic structures on moduli spaces.
method Interpreting complex hyperbolic form as a Weil-Petersson form for punctured spheres.
result Found a new equality between two symplectic forms.
Paper constructs Hopf real hypersurfaces in complex hyperbolic space.
problem Existence of Hopf real hypersurfaces in complex hyperbolic space.
method Twistor spaces and para-quaternionic Kähler structure.
result Unified construction of Hopf real hypersurfaces in complex hyperbolic space.