The paper classifies a space of generalized cusps and its moduli.
problem Classifying the moduli space of generalized cusps.
method Generalized cusp classification, representation theory, and geometric structures.
result The moduli space of generalized cusps is homeomorphic to a subspace of conjugacy classes of representations.
Study characterizes quasi-isometric embeddings of maps from cusped surfaces into moduli space.
problem Characterizing quasi-isometric embeddings of maps from cusped surfaces into moduli space.
method Investigates shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces, considering quasi-isometric embeddings with respect to Teichmüller distance and intrinsic distance.
result Characterizations of quasi-isometric embeddings are solely determined by the map's monodromy under mild conditions.
Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.
problem Infinite volume of moduli spaces of hyperbolic surfaces with cusps.
method Introduce exponential volume form exp(-W)Vol(K,L) where W is a function of hyperbolic areas.
result Exponential volume forms make moduli spaces finite and relevant to open string theory.
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 study finds arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps.
problem Understanding spectral gaps of random hyperbolic surfaces with many cusps.
method Analysis of moduli spaces of hyperbolic surfaces with Weil-Petersson metric.
result Arbitrarily small spectral gaps are observed as the number of cusps grows slower than the genus.
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 …
Study of large-n asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.
problem Understanding the geometry and spectral properties of random hyperbolic surfaces with many cusps.
method Large-n asymptotic analysis, spectral theory, and moduli space volumes. result Linear number of small Laplacian eigenvalues and relative frequency of simple vs. non-simple closed geodesics.
We extend cell decomposition to moduli space of convex projective structures.
problem Cell decomposition of moduli space of convex projective structures.
method Use Fock and Goncharov's A-coordinates and edge-flipping algorithm. result Holonomy groups are semi-arithmetic in many cases.
We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expresse…
The study proves exponential mixing for specific Weil-Petersson flows in exceptional moduli spaces.
problem Analyzing mixing rates of geodesic flows in moduli spaces with cusps.
method New method of analyzing invariant foliations for hyperbolic flows with singularities, using metric changes and flow rescaling.
result Exponential mixing for Weil-Petersson flows in M1,1 and M0,4, contrasting with non-exponential mixing in other moduli spaces. New recursion found for hyperbolic sphere volumes.
problem Volume calculation of hyperbolic sphere moduli spaces.
method Proved a non-linear recursive relation.
result Generalized Zograf's result for conical points and geodesic boundaries.
Classifies topological types of unions of 3-punctured spheres in hyperbolic 3-manifolds.
problem Classifying unions of 3-punctured spheres in hyperbolic 3-manifolds.
method Topological classification based on geometric properties of 3-punctured spheres.
result Special types of unions appear in only one or a few hyperbolic 3-manifolds.
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…
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…
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.
The study constructs pressure form on Margulis spacetimes and proves their infinitesimal rigidity.
problem Understanding the infinitesimal rigidity of Margulis spacetimes.
method Constructing pressure form and studying its properties on the moduli space of Margulis spacetimes.
result Margulis spacetimes are infinitesimally determined by their marked Margulis invariant spectra.
This paper defines the pressure metric on the Moduli space of Margulis spacetimes without cusps and shows that it is positive definite on the constant entropy sections. It also demonstrates an identity regarding the variation of the cross-ratios.
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…
The abstract conjectures a formula for virtual elliptic genera of sheaves on surfaces.
problem Computing virtual elliptic genera of sheaves on surfaces.
method Expressed conjectures in terms of Igusa cusp form, quasi-Jacobi forms, and universal functions.
result Conjectures a formula involving virtual cobordism classes and universal functions.
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.
Study asymptotics of Selberg zeta function on spin moduli space.
problem Asymptotic behavior of Selberg zeta function for degenerating metrics.
method Analyzes logarithmic derivative of Selberg zeta function for spin Dirac operator on compact surfaces.
result Proves asymptotic expansion up to order t4logt. Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end …
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…
Analyzes Quillen norm on determinant line bundle for curves with cusps.
problem Analyzing Quillen norm on determinant line bundle for curves with cusps.
method Studies Quillen norm on determinant line bundle for complex curves with cusps, using analytic torsion.
result Derives explicit formula for curvature current, refining Riemann-Roch-Grothendieck theorem.
Formulae for Masur-Veech volumes derived from intersection numbers of curves.
problem Computing volumes and densities of geodesics and surfaces.
method Intersection numbers of psi-classes and lattice point count.
result Formulae for Masur-Veech volumes as polynomials in intersection numbers.
Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…
The paper defines a path metric on a stable component of polynomial families.
problem Understanding the geometry of polynomial families with parabolic relations.
method Constructing a positive semi-definite pressure form on a bounded stable component of the moduli space.
result The pressure form defines a path metric on the stable component.
A bijection proves a polynomial volume for genus-0 hyperbolic surfaces with boundaries.
problem Proving the Weil-Petersson volume polynomial in boundary lengths for genus-0 surfaces.
method Generalizing a tree bijection to handle geodesic boundaries, extending spine construction.
result Explicit formula for three-point function in Weil-Petersson random surfaces.
It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two …
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.
A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formu…
Mandelbrot set is a closure of the set of zeroes of resultantx(Fn,Fm) for iterated maps Fn(x)=f∘n(x)−x in the moduli space of maps f(x). The wonderful fact is that for a given n all zeroes are not chaotically scattered around the moduli space, but lie on smooth curves, with just a few cusps, located…
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...].
We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', …
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 of IR phases in 3D class R theories linked to non-hyperbolic 3-manifolds.
problem Understanding IR phases of 3D class R theories associated with non-hyperbolic 3-manifolds.
method Analysis of IR phenomena through `exceptional' Dehn fillings and gauging of flavor symmetries.
result 3D class R theories associated with certain atoroidal non-hyperbolic 3-manifolds exhibit supersymmetry enhancement at low energy.
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.
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.
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.
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.
We organize the quantum hyperbolic invariants (QHI) of 3-manifolds into sequences of rational functions indexed by the odd integers N≥3 and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic 3-manifolds M we generalize the QHI and get rati…
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.