Study refines Siegel-Veech constants for abelian differentials.
problem Computing Siegel-Veech constants for abelian differentials.
method Intersection theory and quasimodular forms.
result New identity for Siegel-Veech constants of cylinders.
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48 results for “Siegel-Veech constant”
Study connects geodesic sums to area constant.
problem Understanding geodesic sums in Teichmüller space.
method Relates trimmed sums of twists to area Siegel-Veech constant.
result Established connection between geodesic sums and area constant.
Convergence of Siegel-Veech constants for weakly convergent measures on translation surfaces.
problem Convergence of Siegel-Veech constants for weakly convergent measures on translation surfaces.
method Recurrence result related to Eskin-Masur techniques, measure equidistribution result.
result Convergence of sequences of Siegel-Veech constants associated to Teichmüller curves in genus two.
The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.
problem Calculating area Siegel--Veech constants for affine invariant submanifolds of REL zero.
method Using volumes of the principal boundary strata and intersection theory.
result Proves a conjectural formula for the area Siegel--Veech constant in the case of REL zero.
We extend asymptotic formulas for saddle connections on translation surfaces.
problem Counting saddle connections on translation surfaces with large genus.
method Recursive formulas and asymptotic analysis for all strata and multiplicities.
result Asymptotics for all saddle connections on translation surfaces of growing genus.
Computes constants for specific geometric structures.
problem Calculating constants for specific geometric structures.
method Analyzes saddle connections and Prym eigenforms.
result Computed Siegel-Veech constants for real quadratic orders.
The study predicts large genus behavior of quadratic differential volumes and constants.
problem Predicting large genus behavior of quadratic differential volumes and constants.
method Analyzing conjectures on asymptotic behavior of Masur-Veech volumes and area Siegel-Veech constants.
result Conjectures on large genus asymptotics of quadratic differential volumes and constants.
The study predicts the growth of moduli spaces of Abelian differentials and their constants in large genera.
problem Understanding the asymptotic behavior of moduli spaces and Siegel-Veech constants in large genera.
method Analyzing numerical evidence and recent advances in the field.
result Numerical evidence supports conjectures on the volumes and constants' behavior in large genera.
The paper calculates large genus limits for two types of Siegel-Veech constants.
problem Large genus asymptotics for Siegel-Veech constants in Abelian differentials.
method Combining combinatorial analysis and large genus asymptotics of Masur-Veech volumes.
result The saddle connection and area Siegel-Veech constants converge to specific values as genus grows large.
Computes constants for cyclic covers of translation surfaces.
problem Asymptotic number of cylinders on translation surfaces.
method Topological invariants and number-theoretic properties of degree.
result Ratio of Siegel-Veech constants is independent of degree.
Quasimodular forms help calculate large genus limits of Siegel-Veech constants.
problem Counting torus coverings and their large genus limits.
method Using quasimodular forms and generating functions, connecting geometric definitions with combinatorial counting.
result Proved conjectures on large genus limits of Masur-Veech volumes and Siegel-Veech constants.
Study calculates volumes and constants from intersection theory on abelian differential strata.
problem Calculating volumes and constants from intersection theory on abelian differential strata.
method Intersection numbers on strata with prescribed zeros orders.
result Evaluation of large genus limits and saddle connection Siegel-Veech constants for all strata.
Study proves conjectures about volumes and Siegel-Veech constants for Hodge integrals.
problem Proving conjectures about volumes and Siegel-Veech constants for Hodge integrals.
method Analysis of asymptotic behavior of quasi-modular forms and expressions in terms of Hodge integrals.
result Conjectures about volumes and Siegel-Veech constants for Hodge integrals are proven.
Proves quasimodularity of generating functions for pillowcase covers.
problem Counting Feynman-like graphs associated with quadratic differentials.
method Analyzing decompositions of half-translation surfaces into horizontal cylinders.
result Alternative proof of quasimodularity results and practical method to compute area Siegel-Veech constants.
Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient c…
An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of…
The paper extends Siegel-Veech formula to convex flat cone spheres.
problem No formula exists for flat surfaces with irrational cone angles.
method Defined a generalized Siegel-Veech transform and Siegel-Veech measure.
result The Siegel-Veech measure is absolutely continuous and piecewise real analytic.
Study on wind-tree models yields formulas for periodic trajectories.
problem Counting periodic trajectories in wind-tree models.
method Asymptotic formulas and explicit computation of Siegel-Veech constants.
result Asymptotic formulas for closed billiard trajectories in wind-tree models.
We calculate volumes of quadratic differentials using topological recursion.
problem Calculating volumes of quadratic differentials on curves.
method Topological recursion and geometric recursion applied to hyperbolic lengths of multicurves.
result Formula for constant terms of polynomials in terms of stable graphs.
We present an explicit formula relating volumes of strata of meromorphicquadratic differentials with at most simple poles on Riemann surfacesand counting functions of the number of flat cylinders filled by closedgeodesics in associated flat metric with singularities. This generalizes the resultof Athreya, Eskin and Zor…
A higher moment formula for the Siegel--Veech transform over quotients by Hecke triangle groupsmath.GT
Formula calculates higher moments of Siegel-Veech transform over Hecke triangle groups.
problem Computing higher moments of Siegel-Veech transform over specific groups.
method Geometric results and linear algebra to create integration formulas.
result Explicit integration formulas for densities of vector orbits.
The paper calculates volumes of Abelian differential strata in large genus asymptotics.
problem Calculating volumes of Abelian differential strata in large genus asymptotics.
method Combinatorial analysis of Eskin-Okounkov's algorithm to evaluate Masur-Veech volumes.
result The volume of a stratum indexed by a partition is (4 + o(1)) * prod(m_i + 1)^(-1) as 2g - 2 = sum(m_i) tends to infinity.
Formula for Masur-Veech volumes in quadratic differentials with odd zeros.
problem Calculating volumes of specific quadratic differential strata.
method Intersection theory, topological recursion, Hodge integrals.
result Conjectural formula for volumes proved for odd zero orders.
Proves quasimodularity of generating functions for torus covers.
problem Counting torus covers with and without Siegel-Veech weight.
method Analyzing decompositions of flat surfaces into horizontal cylinders, using quasi-elliptic functions.
result Quasimodularity arises as contour integral of quasi-elliptic functions.
We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that…
The paper decomposes spectral functions on marked tori strata.
problem Decomposing square-integrable functions on strata of differentials.
method Spectral decomposition and analysis of differential operators.
result The continuous spectrum of the foliated Laplacian is larger than Siegel-Veech transforms.
For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. I…
Consider the 1-dimensional Hurwitz space parameterizing covers of P^1 branched at four points. We study its intersection with divisor classes on the moduli space of curves. As an application, we calculate the slope of the Teichmuller curve parameterizing square-tiled cyclic covers and recover the sum of its Lyapunov ex…
The paper calculates large genus limits for quadratic differential volumes and constants.
problem Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials.
method Combining recursive relations (Virasoro constraints) and asymmetric simple random walk jump probabilities.
result Confirm predictions about Masur-Veech volumes and area Siegel-Veech constants.
Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.
problem Analyzing flat metrics from right regular prisms.
method Viewing prisms as n-differentials and analyzing unfoldings, proving translation coverings to hyperelliptic surfaces.
result Non-lattice surfaces admit translation coverings to hyperelliptic surfaces, allowing explicit computation of orbit closures and counting problems.
Study orbits of discrete lattice actions on the plane, derive new results for Veech surfaces.
problem Count pairs of holonomy vectors in Veech surfaces with bounded parameters.
method Siegel-Veech-type integral formula for averages of pairs of orbits.
result Upper bounds on pairs of holonomy vectors in Veech surfaces with bounded parameters.
We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stra…
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…
Formulae for Masur-Veech volumes and frequencies of geodesics derived from intersection numbers.
problem Calculating volumes and frequencies of geodesics in moduli spaces.
method Lattice point counts and intersection numbers of ψ-classes, with explicit rational coefficients.
result Formulae for Masur-Veech volumes and frequencies of simple closed geodesics.
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.
Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.
problem Identifying Clairaut constants from Fermat constants for specific geodesics.
method Analytical proof for a specific class of geodesics on a surface of revolution.
result Fermat constants do not fully determine Clairaut constants for some geodesics, except for a standard sphere.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
The paper classifies hypersurfaces in with constant curvature.
problem Classifying hypersurfaces in with constant sectional curvature.
method Analyzing the geometry of and constructing specific examples.
result Examples of hypersurfaces in with non-constant product angle function.
Study on biconservative hypersurfaces with constant scalar curvature in space forms.
problem Characterize biconservative hypersurfaces with constant scalar curvature in space forms.
method Analyzing biconservative hypersurfaces in space forms , proving properties and finding specific examples.
result Proves that biconservative hypersurfaces with constant scalar curvature in have constant mean curvature, and in , they are either rotational or constant mean curvature.
Study constant angle surfaces in 4D Minkowski space, proving their properties.
problem Characterize surfaces in 4D Minkowski space with constant angle between tangent planes.
method Define complex angle, prove curvature properties, use PDE methods, analyze special cases.
result Constant angle surfaces have vanishing Gauss and normal curvatures; not complete for .
We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…
The study classifies surfaces with constant slope in 4D space.
problem Classifying surfaces with constant slope in higher dimensions.
method Analyzing generalized constant ratio surfaces in Euclidean 4-space.
result A classification of constant slope surfaces.
The paper studies curves of constant-ratio in pseudo-Galilean space.
problem Characterizing curves of constant-ratio in pseudo-Galilean space.
method Analyzing spacelike curves with constant-ratio in terms of curvature functions.
result Characterization of special curves of constant-ratio in pseudo-Galilean space.
Study CR Yamabe constant and CR structures on manifolds.
problem Understanding CR Yamabe constant and its role in CR geometry.
method Developed integral formulae and constructed families of CR structures.
result Found an infinite family of CR structures with varying CR Yamabe constants.
The Cheeger constant increases under Ricci flow on spheres.
problem Behavior of the Cheeger constant under Ricci flow.
method Evolution identities for parallel curves and viscosity formulation of .
result The Cheeger constant is non-decreasing under Ricci flow on surfaces diffeomorphic to .
Study on higher-order Escobar constants for planar domains.
problem Understanding Escobar constants for planar domains of higher order.
method Investigation of higher-order Escobar constants on bounded planar domains .
result Escobar constants for the unit disk and a family of polygons are provided.
The paper defines new constants for -Laplacian on manifolds.
problem Bounding eigenvalues of the -Laplacian on compact manifolds.
method Introducing Steklov and Neumann isocapacitary constants.
result Two-sided bounds for -Sobolev constants and eigenvalues.
Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.