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.
Several new combinatorial descriptions of closed 4-manifolds have recently been introduced in the study of smooth maps from 4-manifolds to surfaces. These descriptions consist of simple closed curves in a closed, orientable surface and these curves appear as so called vanishing sets of corresponding maps. In the presen…
4 flat 3-manifolds realized in hyperbolic 4-space.
problem Realizing flat 3-manifolds in hyperbolic 4-space.
method Using finite-volume hyperbolic 4-manifolds with cusp sections.
result 6 closed orientable flat 3-manifolds realized as cusp sections.
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.
Study on shortest geodesics crossing multiple times on hyperbolic surfaces with cusps.
problem Finding shortest geodesics crossing multiple times on hyperbolic surfaces.
method Investigates closed geodesics on hyperbolic surfaces with at least one cusp, focusing on minimal length and self-intersection numbers.
result For large enough k, self-intersection numbers are exactly k for geodesics crossing at least k times. Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…
New hyperbolic 3-manifolds with multiple cusps are found that sound the same but look different.
problem Finding hyperbolic 3-manifolds with multiple cusps that are isospectral but not isometric.
method Used Sunada's method and the Strong Approximation Theorem of Nori and Weisfeiler.
result Constructed hyperbolic 3-manifolds with multiple cusps that are isospectral but not isometric.
Currents on cusped hyperbolic surfaces have a denseness property similar to compact surfaces.
problem Proving denseness of rational currents on cusped hyperbolic surfaces.
method Using geodesic currents and subset currents, proving denseness through examples and continuous extension.
result Denseness of rational currents on cusped hyperbolic surfaces, including geodesics connecting cusps.
A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodifferential operators on the fibres (with …
Survey of analytic and geometric results on fibred cusp spaces.
problem Analyzing a class of non-compact Riemannian manifolds with cusp singularities.
method Careful microlocal analysis of the resolvent and heat kernel.
result Main theorems on spectral geometry, including analytic torsion and index theory.
We show that the number of isometry classes of cusped hyperbolic 3-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.
Calderón projector extended to fibred cusp operators.
problem Extending Calderón projector to non-compact manifolds with special fibred structure.
method φ-pseudodifferential calculus by Mazzeo and Melrose.
result Calderón projector for fibred cusp operators is a pseudodifferential operator.
We construct infinitely many examples of pairs of isospectral but non-isometric 1-cusped hyperbolic 3-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada's method in the cusped setting, and so in addition our pairs are fin…
It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic 3-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete cl…
Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.
problem Understanding length statistics of geodesics on random hyperbolic surfaces with cusps.
method Recursion formula for tight Weil-Petersson volumes and generalization of Mirzakhani's integration formula.
result Recovery of Poisson point process in large genus limit for length statistics of tight geodesics.
We show that the set of cusp shapes of hyperbolic tunnel number one manifolds is dense in the Teichmuller space of the torus. A similar result holds for tunnel number n manifolds. As a consequence, for fixed n, there are infinitely many hyperbolic tunnel number n manifolds with at most one exceptional Dehn filling. Thi…
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 on geodesics and eigenvalues on random hyperbolic surfaces with cusps.
problem Counting short geodesics and small eigenvalues on random hyperbolic surfaces.
method Rescaling and convergence to a Poisson point process.
result The probability of having at least k=o(n) arbitrarily small eigenvalues tends to 1 as no∞. In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow…
We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of R-circles, is connected, and contains a Hopf link with three components; we also show that the fundamental group of its complement in S3 …
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.
In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n-manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n+1)-orbifold is dense in the space of similarity classes of flat metri…
We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing d…
The paper classifies different types of cusps on plane curves.
problem Investigating various types of cusps on plane curves.
method Examining criteria for (n,n+1) cusps with differential conditions and relations to evolutes of fronts. result Complete classifications for (4,5)-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.
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.
Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.
problem Enhance graph representation learning by leveraging spectral and curvature signals.
method Proposes Spectro-Riemannian Graph Neural Networks (CUSP) that combines spectral and curvature insights.
result Empirical evaluation shows CUSP outperforms state-of-the-art models by up to 5.3%.
Study of lambda lengths in figure eight knot complement using Eisenstein integers.
problem Determining lambda lengths in the figure eight knot complement.
method Using hyperbolic geometry and spinors, mapping lambda lengths to Eisenstein integers.
result Lambda lengths are precisely the Eisenstein integers, up to multiplication by a unit.
Study projective deformations of hyperbolic 3-orbifolds with turnover ends.
problem Deformations of hyperbolic 3-orbifolds with turnover ends in projective geometry.
method Projective deformations of hyperbolic 3-orbifolds with turnover ends, focusing on totally geodesic generalized cusps.
result Turnover funnels remain totally geodesic and the deformed projective 3-orbifold remains properly convex.
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 PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
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.
Research shows how certain flat structures behave in specific convex domains.
problem Understanding the behavior of codimension-1 simplices in divisible convex domains.
method Analyzes the set of codimension-1 flats and their images in quotient manifolds.
result The set of codimension-1 flats forms a finite collection of disjoint virtual tori, leading to cusped convex projective manifolds.
We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling w…
Constructs stable maps from 3-manifolds to surfaces without cusps.
problem Creating stable maps from 3-manifolds to surfaces without problematic points.
method Visual construction of stable maps with specific properties.
result Obtains stable maps with no cusps and specific fiber structures.
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.
In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index…
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
Study of Eisenstein series linked to hyperbolic cusps.
problem Understanding Eisenstein series associated with hyperbolic cusps.
method Analyzing cohomology classes and intertwining operators.
result Different cusps correspond to linearly independent cohomology classes.
This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.
problem Establishing exponential mixing for frame flows on geometrically finite hyperbolic manifolds with cusps.
method Symbolic coding of geodesic flow, Dolgopyat's method, large deviation property, combinatorics of cusp excursions, renewal theorem.
result Frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing.
We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. It follows that a sufficiently small deformation of the holonomy of a finite volume strictly convex real projective manifold is the holonomy of some nearby projective structure with r…
We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume vm=4π2/3 and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume 2⋅vm and one cusp. It has lowest volume among…
The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour …
Study Dirac operator on cusped hyperbolic manifolds, finding spectrum properties.
problem Investigate the Dirac operator's spectrum on hyperbolic manifolds with cusps.
method Analyze spin structures on finite-volume hyperbolic n-manifolds, focusing on cusps.
result Discovered examples where Dirac operator's spectrum is R in some dimensions and discrete in others.
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.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Let M be a two cusped hyperbolic 3-manifold and let M(r) be the result of r Dehn filling of a fixed cusp of M. We study canonical components of the SL(2,C) character varieties of M(r). We show that the gonality of these sets is bounded, independent of the filling parameter. We also obtain bounds, depending on r, for th…
Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.
problem Localization of Bergman kernels on Kähler manifolds with complex hyperbolic cusps.
method Revisiting Tian's peak section method, applying to Kähler-Einstein metrics and quotients of complex balls.
result Partial localization result for Poincaré type cusps.