Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.
problem Counting and equidistribution of cusped Hitchin representations.
method Renewal theorem of Kesseböhmer and Kombrink applied to count and equidistribute.
result Entropy gaps at infinity allow for counting and equidistribution results.
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.
Counts arcs in surfaces, proving convergence of geodesic currents.
problem Counting arcs of the same type in compact surfaces and related geometries.
method Derives convergence of geodesic currents to prove arc counts.
result Proves convergence of geodesic currents, leading to arc counting results.
The study counts cusps on finite volume Riemannian manifolds using volume and decay estimates.
problem Counting cusps on finite volume Riemannian manifolds.
method Volume and decay estimates, nonlinear theory of p-Laplacian, volume comparison theorems.
result An upper bound on the number of cusps based on the volume of the manifold.
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.
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 …
New math shows many 3D hyperbolic shapes can fit together.
problem Counting specific 3D shapes that fit together.
method Examined both arithmetic and non-arithmetic shapes, focusing on their volume.
result The number of such shapes grows super-exponentially with volume.
The study counts cusps in deformed complex polynomials near the origin.
problem Understanding singularities in deformed complex polynomials.
method Analyzes deformations of complex polynomials to real maps with fold and cusp singularities.
result Calculates the number of cusps in a small neighborhood of the origin.
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. We study the asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M. In the appendix, due to K. Bel…
The entropy of minimal surfaces is minimized in hyperbolic manifolds.
problem Counting essential minimal surfaces in closed negatively curved manifolds.
method Defining minimal surface entropy and computing it for various spaces.
result Minimal surface entropy is minimized in hyperbolic manifolds.
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to L6g−6+2(n+p). We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the cor…
Efficient cobordisms show minimal signature values on certain links.
problem Finding minimal signature values for specific link types.
method Constructing topological cobordisms between torus links and connected sums of trefoil knots.
result The signature invariant σω at ω=ζ6 takes minimal values on torus links. Study counts geodesics on modular surface, linking to necklace counting.
problem Counting geodesics on modular surface with specific winding numbers.
method Asymptotic expansion, generating function analysis, correspondence to necklace counting.
result Obtained asymptotic growth rate of m low-lying geodesics in terms of word length.
Counting subgroups of a surface using convex core lengths.
problem Counting conjugacy classes of subgroups of fundamental groups of surfaces.
method Using half the sum of the lengths of the boundaries of the convex core of a subgroup.
result The number of conjugacy classes of subgroups is asymptotic to cL6g−6+2r. Study growth rates in ball quotients and maps onto integer groups.
problem Growth rates of first betti numbers and cusp counts in ball quotients.
method Analysis of finite-sheeted coverings and arithmetic lattices.
result Explicit example of lattice with homomorphism to Z with finitely generated kernel.
Improved bounds on a function linking geodesics to moduli spaces.
problem Understanding geodesics on hyperbolic surfaces.
method Analyzing the Mirzakhani function and its behavior near cusps.
result The function is square-integrable with respect to the Weil-Petersson volume form.
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.
We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric gWP on Mγ, the Riemann moduli space of surfaces of genus γ>1. This space has a singular compactification with respect to gWP, and this metric has crossing…
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 n≥3 we show that the numerical dimension of the canonical divisor of a smooth n-dimensional compactification is always …
The paper studies the growth of lengths of curves on hyperbolic surfaces.
problem Asymptotic growth of lengths of closed curves on hyperbolic surfaces.
method Ergodic properties of the earthquake flow on moduli space.
result Asymptotics of the number of closed curves of length ≤ L on a hyperbolic surface.
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.
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.
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.
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.
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.
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.
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 k=2. New approach to extremal hyperbolic surfaces using NEC groups.
problem Structural description of extremal hyperbolic surfaces.
method Uniformization by NEC groups for surfaces with cusps and/or geodesic boundary.
result Full description of automorphism groups of extremal surfaces.
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.
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.
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.
Computes cusp cobordism groups for Morse functions on manifolds.
problem Understanding the cusp cobordism groups of Morse functions.
method Employed Levine's cusp elimination technique and created pairs of cusps along fold lines.
result Both unoriented and oriented cusp cobordism groups are cyclic of order two in even dimensions and infinite order in odd dimensions.
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.
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.
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…
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.
This paper studies the waist size of cusps in hyperbolic 3-manifolds, proving unique smallest sizes for specific manifolds.
problem Determining the smallest waist sizes of cusps in hyperbolic 3-manifolds.
method Analyzing the shortest nontrivial curves generated by parabolic isometries in maximal cusp boundaries.
result The next two smallest waist sizes are realized uniquely for specific manifolds.
Study on cusp singularities in Kähler-Einstein metrics.
problem Analyzing cusp singularities in Kähler-Einstein metrics.
method Continuity method applied to varieties with cusp singularities.
result Investigation of differential and algebro-geometric properties of the limit.
Geodesic tetrahedra found for Platonic cusped manifolds.
problem Finding geodesic decompositions for Platonic cusped manifolds.
method Decomposing Platonic cusped hyperbolic manifolds into geodesic ideal tetrahedra.
result Geodesic triangulations exist for Platonic cusped 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.
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…