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.
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48 results for “cusp compactifications”
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…
We classify the smallest finite volume complex hyperbolic surfaces with cusps which admit smooth toroidal compactifications and which are not birational to a bi-elliptic surface. Remarkably, there is only one such surface which appears to be the compactification of a Picard modular surface.
We prove a bound relating the volume of a curve near a cusp in a hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. There are a number of consequences: we show that for …
Constructs ball quotient compactifications and studies lattices in PU(2,1).
problem Studies lattices in PU(2,1) with specific properties.
method Constructs ball quotient compactifications and analyzes associated lattices.
result Shows existence of lattices in PU(2,1) with infinite abelianization and finitely generated commutator subgroup.
Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theoremmath.DG
Study of Quillen metric on Riemann surfaces with cusps and compactification.
problem Behavior of Quillen metric on Riemann surfaces with cusps.
method Continuous extension of Quillen metric over singular curves and explicit universal constant.
result Compatibility of Quillen metric with clutching morphisms and analytic torsion.
The paper studies positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
problem Positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
method Analyzes intrinsic positivity properties of cotangent bundles using toroidal compactifications and ample line bundles.
result The cotangent bundle is ample modulo the boundary divisor for sufficiently small rational numbers.
We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebro-geometric methods using Mumford's work on toroidal compactifications and its generalization due to N. Mok and W.-K. To, we get a bound which is considerably better than those obtained previousl…
In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be nef if the dimension is greater or equal to three. Moreover, if we show that the numerical dimension of the canonical divisor of a smooth -dimensional compactification is always …
Formula connects analytic and Reidemeister torsions for hyperbolic manifolds.
problem Relating analytic and Reidemeister torsions for hyperbolic manifolds.
method Derives a formula relating analytic torsion to Reidemeister torsion of compactified manifolds.
result Formula connects analytic and Reidemeister torsions for hyperbolic manifolds.
We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We…
In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to the combinatorics of the graph. We then compactify this surface by adding finit…
Generalizes Riemann-Roch isometry to non-compact orbifold settings.
problem Applying Riemann-Roch isometry to cusps and elliptic fixed points of Riemann surfaces.
method Surgery techniques, Mayer-Vietoris formulae, Quillen metrics, Selberg zeta function.
result Determinant of cohomology related to explicit metrics and line bundles.
Compactifies geodesic flows on hyperbolic surfaces, revealing attractive circles at infinity.
problem Geodesic flows on non-compact hyperbolic surfaces without cusps.
method Constructs a geometrical compactification using one-dimensional distributions tangent to stable and unstable horocycles.
result Existence of attractive circles at infinity in the compactified flow.
For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems we define the Reidemeister torsion of the Borel-Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we rela…
We describe typical degenerations of quadratic differentials thus describing ``generic cusps'' of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from ``generic'' degenerations is often negligible in problems involving …
New PL invariant classifies K3 surface degenerations.
problem Classifying type II degenerations of K3 surfaces.
method Explicit PL convex function from interval, differential geometric viewpoint.
result Function classifies degenerations into combinatorial types.
We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric on , the Riemann moduli space of surfaces of genus . This space has a singular compactification with respect to , and this metric has crossing…
The paper studies metrics on complex hyperbolic manifolds with cusps and their properties.
problem Analyzing metrics on complex hyperbolic manifolds with cusps and their properties.
method Using singular metrics and curvature conditions to study the bigness and nefness of tangent and cotangent bundles.
result Effective ramification orders for covers to have big cotangent bundles and high enough ramification makes the tangent bundle nef.
Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.
problem Exponential growth of torsion in the cohomology of arithmetic groups.
method Analytic torsion and Reidemeister torsion, applied to fibered cusp ends of manifolds.
result Exponential growth of torsion in the cohomology of arithmetic groups.
In this paper, we consider solutions and spectral functions of M-theory from Milne spaces with extra free dimensions. Conformal deformations to the metric associated with the real hyperbolic space forms are derived. For the three-dimensional case, the orbifold identifications …
New invariant cusp crossing density shows cusp densities of hyperbolic knots and links are dense.
problem Densities of cusp invariants for hyperbolic knots and links.
method Defined cusp crossing density as ratio of cusp volume to crossing number; showed densities are dense.
result Cusp crossing density for links is bounded above by 3.1263... and is dense in [0, 2.120...].
4-manifolds show every flat 3-manifold as cusp sections.
problem Realizing flat 3-manifolds as cusp sections of hyperbolic 4-manifolds.
method Transitive action on cusps, dense flat metrics realization.
result Existence of many cusp-transitive 4-manifolds.
Study shows Satake compactifications as horofunctions with polyhedral metrics.
problem Understanding horofunction compactifications of symmetric spaces.
method Investigated invariant Finsler metrics and polyhedral metrics for compactification.
result Generalized Satake compactifications realized as horofunction compactifications.
Characterizes holonomies of convex projective cusps.
problem Understanding holonomies in strictly convex projective geometry.
method Complete characterization of holonomies for strictly convex and round cusps, building families of generalized cusps.
result Produces the first example of generalized cusps with non-virtually nilpotent fundamental group.
Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
Proves methods for creating convex projective 3-manifolds with cusps.
problem Creating convex projective 3-manifolds with generalized cusps.
method Properly convex deformations of hyperbolic structures, controlling cusp types.
result First known example of a 1-cusped hyperbolic 3-manifold with a type 2 cusp.
We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…
The paper classifies different types of cusps on plane curves.
problem Investigating various types of cusps on plane curves.
method Examining criteria for cusps with differential conditions and relations to evolutes of fronts.
result Complete classifications for -cusps.
The paper characterizes links in 3D from divides with cusps.
problem Characterizing links in 3D from divides with cusps.
method Defines and characterizes divides with cusps and their associated links.
result Every strongly invertible link and 2-periodic link can be described as the link of a divide with cusps.
Extends techniques to show existence of all cusp types in convex projective manifolds.
problem Existence of all cusp types in convex projective manifolds.
method Extension of techniques by Ballas-Marquis.
result Existence of all cusp types in all dimensions except diagonalizable.
The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.
problem Understanding cusp types in hyperbolic 4-manifolds and their commensurability.
method Criteria for commensurability classes containing specific cusp types.
result Infinitely many examples of commensurability classes without certain cusp types.
A new compactification of outer space is an absolute retract.
problem Compactification of outer space for -actions on -trees.
method Defining a new compactification as a blow-up of the classical compactification , assigning orientations to arcs.
result The new compactification is an absolute retract.
The paper constructs examples of compactifications for Einstein metrics.
problem Compactifying Einstein metrics with specific properties.
method Constructing projective and --projective compactifications.
result Neutral signature Einstein metrics can be compactified canonically.
The paper classifies cusp types in hyperbolic knot complements.
problem Classifying cusp types in hyperbolic knot complements.
method Analyzing quotients of hyperbolic knot complements.
result All cusp types arise in the quotients of link complements.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in to .
result Cusped Borel Anosov representations with specific properties are Hitchin representations.
Extends harmonic maps compactification to punctured Riemann surfaces.
problem Compactifying Teichmüller spaces for punctured Riemann surfaces.
method Using harmonic maps rays to extend compactification.
result The compactification still coincides with Thurston's compactification.
Cusped hyperbolic 3-manifolds can have up to 4 cusps under certain group actions.
problem Understanding the maximum number of cusps in hyperbolic 3-manifolds under group actions.
method Analyzing the isometry group actions on cusps of hyperbolic 3-manifolds.
result Constructing a family of manifolds with no upper bound on the number of cusps for .
Paper relates new compactification to classical moduli space.
problem Relating new compactification to classical moduli space.
method Defining a continuous finite-to-one surjection between compactifications.
result Uniform upper bound on cardinalities of fibers.
Study shows rigidity in cusp-decomposable manifolds' geometry.
problem Understanding the geometry of cusp-decomposable manifolds.
method Examined large scale geometry and quasi-isometries.
result Proved quasi-isometric rigidity for fundamental groups.
Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.
problem Identifying hyperbolic 3-manifolds with minimal volume and maximal cusps.
method Maximal cusp volume classification and analysis of low-volume manifolds.
result Figure-8 knot complement is unique in certain volume and filling categories.
The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
problem Characterizing the horoboundary of Teichmüller space.
method Using the relationship between the horofunction and visual compactifications of Teichmüller spaces.
result The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
Existence of Kähler-Einstein metrics on compactifications of Lie groups.
problem Existence of Kähler-Einstein metrics on -Fano compactifications of Lie groups.
method Proving existence through compactifications of Lie groups.
result Classification of -Fano compactifications of with Kähler-Einstein metrics.
Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…
Study of isometries on hyperbolic 3-manifold cusps.
problem Understanding transitivity in hyperbolic 3-manifold actions.
method Analyzing multiply transitive actions of isometries on cusps.
result Proved a conjecture about the maximum transitivity and upper bounds on cusps.
Semi-toric and toroidal compactifications as log minimal models, and applications to weak K-modulimath.AG
Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
problem Characterizing and applying toroidal and semi-toric compactifications to weak K-moduli.
method Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
result Different proof of a theorem of Alexeev-Engel on weak K-moduli compactifications.
Four hyperbolic 24-cell 4-manifolds with one cusp are identified.
problem Identifying hyperbolic 24-cell 4-manifolds with one cusp.
method Analyzing hyperbolic geometry and isometry.
result Found four one-cusped hyperbolic 4-manifolds of minimum volume.
Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactificationsmath.GT
The paper explores ball quotient compactifications and their properties.
problem Smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres.
method Use totally geodesic punctured spheres to prove ampleness of for .
result First examples of bielliptic ball quotient compactifications are produced.