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.
Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.
problem Characterizing dense points in the Bers boundary of Teichmüller space for infinite-type surfaces.
method Quasiconformal deformations and analytic conditions.
result Maximal cusps cannot approach certain points on the Bers boundary.
Classifies worst approximable rational numbers using hyperbolic geometry.
problem Classifying worst approximable rational numbers.
method Geometric approach using hyperbolic geometry.
result Clarifies distinction between two types of worst approximable rationals.
Let f be a 1-variable complex polynomial such that f has a singularity at the origin. In the present paper, we show that there exists a deformation of f which has only fold singularities and cusps as singularities of a real polynomial map from the plane to the plane. We then calculate the number of cusps of a deformati…
A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighb…
New tiling algorithm for hyperbolic 3-manifolds, characterizing cusp areas.
problem Characterizing and computing maximal cusp areas in hyperbolic 3-manifolds.
method Developed a new tiling algorithm and provided simpler expressions for distances.
result Completely characterized the space of cusp neighborhoods and found the Epstein-Penner decomposition.
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a…
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.
We prove that the envelope of meromorphy of any imbedded symplectic sphere in CP2 coincides with the whole CP2. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as adjunction formula, smoothness of moduli space in the neighborhood of a cusp-curve…
An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a h…
Study on scattering geodesics on modular surface and their sojourn times.
problem Distribution of scattering geodesics and their sojourn times on modular surface.
method Analysis of scattering geodesics in modular surface, establishing connection to prime divisors in arithmetic progression.
result Established a connection between scattering geodesics and prime divisors in arithmetic progression.
Given for instance a finite volume negatively curved Riemannian manifold M, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of M and their linear divergence rates under the geodesic flow. As…
We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to natura…
Extending BTZ models to complete hyperbolic surfaces.
problem Extending BTZ models to complete hyperbolic surfaces.
method Proving a parametrization result for globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
result The tangent bundle of the Teichmüller space parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
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.
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.
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.
Research examines correlations of complex logarithms of lattice points, showing level repulsion and Poissonian behavior.
problem Analyzing correlations of complex logarithms of lattice points.
method Proving existence of pair correlation functions and examining behavior at various scalings.
result Level repulsion observed at linear scaling, Poissonian behavior at sublinear scalings.
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.
The study examines correlations of logarithms of integers at different scalings.
problem Analyzing pair correlations of logarithms of integers at various scalings.
method Examined correlations of logarithms of positive integers at different scalings, proving the existence of pair correlation functions.
result Level repulsion at linear scaling, total loss of mass at superlinear scalings, and Poissonian behavior at sublinear scalings.
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.
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…
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.
Applying logarithmic transformations along 2-tori, we construct a generalized complex structure J_n with n type changing luci for every n≥0 on genus 1-Lefschetz fibrations with a cusp neighborhood, which include elliptic surfaces with non-zero euler characteristic. Applying a technique of broken Lefschetz fibrati…
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.
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…
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.
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.
Recent work of Ballas, Cooper, and Leitner identifies (n+1) types of n-dimensional convex projective cusps, one of which is the standard hyperbolic cusp. Work of Ballas-Marquis, and Ballas-Danciger-Lee give examples of these exotic (non-hyperbolic) type cusps in dimension 3. Here an extension of the techniques of…
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…
The study shows how to embed cusp-decomposable manifolds quasi-isometrically.
problem Embedding cusp-decomposable manifolds quasi-isometrically.
method Using properties of the electric space of the universal cover, we show quasi-isometric embeddings.
result Isomorphisms between fundamental groups of higher graph manifolds preserve the decomposition into pieces.
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.
Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.
problem Finding the minimum volume of a 3-cusped orientable hyperbolic 3-manifold.
method Using guts in sutured and pared manifolds.
result The volume of a 3-cusped orientable hyperbolic 3-manifold is at least 5.49... = 6 × Catalan's constant.
Study pressure metrics for cusped Hitchin representations.
problem Characterize cusped Hitchin representations of Fuchsian groups.
method Develop pressure metrics associated to fundamental weights and roots.
result New pressure metrics for Hilbert length when d=3. We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also …
The study analyzes spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.
problem Analyzing spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.
method Equivariant index theorem for Dirac operators on manifolds with φ-cusps under conditions on φ. result The cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero.
Proves existence of circle patterns on surfaces with cusps.
problem Existence of circle patterns with prescribed angles on surfaces with cusps.
method Introduced combinatorial Ricci and Calabi flows to prove longtime existence and convergence.
result Existence of generalized circle patterns with prescribed angles on surfaces with cusps.
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.
The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In …
In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometr…
As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space Hn has at least one cusp for n≥5. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least th…
The study identifies flat manifolds with unique cusp cross-sections in arithmetic hyperbolic manifolds.
problem Characterizing flat manifolds that have unique cusp cross-sections in arithmetic hyperbolic manifolds.
method Algebraic characterization of cusp cross-sections in arithmetic hyperbolic manifolds.
result Construction of flat manifolds with unique cusp cross-sections and proof of their existence in all dimensions n≥32.