Quadratic differentials on punctured surfaces link foliations and metric graphs.
problem Understanding the structure of quadratic differentials on punctured surfaces.
method Introducing asymptotic directions and analyzing foliations and metric graphs.
result A unique meromorphic quadratic differential can be constructed for any prescribed horizontal foliation.
The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.
problem Understanding singularities and deformations in meromorphic connections and quadratic differentials.
method Local formal invariants and jets of meromorphic quadratic differentials, universal isomonodromic deformation, unfolded Stokes phenomenon, horizontal and vertical foliations.
result Establishes a correspondence between local formal invariants and jets of meromorphic quadratic differentials, describing parameter spaces and moduli spaces.
Study uses graph techniques to understand meromorphic quadratic differential strata.
problem Understanding the topology of meromorphic quadratic differential strata.
method Exchange graph techniques to study fundamental groups; generalizes relations for mixed-angulations.
result Explicit presentations of fundamental groups in genus-zero case with four singularities.
Transforms study meromorphically isothermic surfaces near singular points.
problem Analyzing meromorphically isothermic surfaces near singular points.
method Apply Darboux and Calapso transformations to patches of surfaces.
result Transformed patches exhibit continuity around singular points.
This paper defines families of quadratic differentials for a surface.
problem Understanding spaces of quadratic differentials on surfaces.
method Defining families of quadratic differentials and their parametrization.
result A complete account of the theory needed for spaces of quadratic differentials.
The study describes how quadratic differentials influence foliations on Riemann surfaces.
problem Understanding how quadratic differentials affect foliations on Riemann surfaces.
method Analyzing infinite-energy harmonic maps from Riemann surfaces to R-trees with prescribed behavior at poles.
result Any measured foliation is uniquely realized by a meromorphic quadratic differential with prescribed principal parts at poles.
New geometric Joyce structures on moduli spaces of quadratic differentials.
problem Constructing Joyce structures on moduli spaces of quadratic differentials.
method Isomonodromic deformations of second-order linear ODEs with rational potential.
result Construction of Joyce structures on moduli spaces of quadratic differentials.
We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their bounda…
Quadratic differentials induce spiralling foliations on Riemann surfaces.
problem Understanding the structure of foliations induced by quadratic differentials.
method Introduced a space of measured foliations and used harmonic maps to real trees.
result Any measured foliation is realized by a quadratic differential with second order poles at marked points.
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 volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quad…
We prove a formula for the determinant of Laplacian on an arbitrary compact polyhedral surface of genus one. This formula generalizes the well-known Ray-Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic surface is also cons…
Paper defines quasi-Strebel structures for meromorphic k-differentials and proves their existence.
problem Existence of quasi-Strebel structures for meromorphic k-differentials.
method Introduced quasi-Strebel structures and proved their existence for meromorphic k-differentials.
result Every differential of even order k > 2 satisfying certain conditions admits a quasi-Strebel structure.
Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low genera the strata of meromorphic quadratic …
Study special circle bundles over moduli spaces of quadratic differentials.
problem Analyse of circle bundles over specific moduli spaces of quadratic differentials.
method Formalism of the Bergman tau-function applied to combinatorial models.
result Sections of circle bundles over moduli spaces are given by the argument of the modular discriminant.
The paper constructs harmonic maps from punctured surfaces to the hyperbolic plane.
problem Harmonic maps from punctured surfaces to hyperbolic planes.
method Constructing polynomial growth harmonic maps from punctured Riemann surfaces to regular polygons in hyperbolic plane.
result Established uniqueness of harmonic maps within a class of maps differing by exponentially decaying variations.
We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor-Bendix…
Study meromorphic k-differentials on Riemann surfaces, focusing on their singularities.
problem Characterize the local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeroes and poles, and k-residues at poles, for different genera.
result For genus g ≥ 2, every expected tuple of k-residues appears as actual residues.
We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on CP1 and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenki…
This paper shows how to create quadratic differentials with any given singularities.
problem Creating quadratic differentials with prescribed singularities.
method Using the flat metric induced by the differentials, the authors classify and construct quadratic differentials with specific singularities.
result Every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface, with exceptions in genera zero and one.
The paper proves a grafting theorem for meromorphic projective structures and shows the monodromy map is a local homeomorphism.
problem Understanding projective structures on Riemann surfaces with poles.
method Proves a grafting theorem involving crowned hyperbolic surfaces and uses the monodromy map to a decorated character variety.
result The monodromy map to the decorated character variety is a local homeomorphism.
The study classifies Veech groups of flat surfaces with poles.
problem Classifying Veech groups of flat surfaces with poles.
method Classification through orbit closure and determination of Veech groups.
result Characterization of surfaces with closed orbits and determination of Veech groups.
Extends Teichmüller space parametrization using poles of higher order.
problem Parametrize Teichmüller space of crowned hyperbolic surfaces.
method Use meromorphic quadratic differentials with higher order poles to parametrize.
result Existence of harmonic map from punctured Riemann surface to crowned hyperbolic surface.
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 …
Let X be some Riemann surface, and let omega be a meromorphic quadratic differential form on X, that is, omega can be written in local coordinates as f(z) dz^2, for some meromorphic function f. We say that a curve gamma is part of a horizontal leaf of omega if for each t in I, we have that f(gamma(t)) (gamma'(t))^2 is …
New parameterizations for anti-de Sitter structures on surfaces.
problem Understanding deformation spaces of anti-de Sitter structures.
method Introducing regular globally hyperbolic structures and providing two parameterizations.
result Two parameterizations of deformation space of anti-de Sitter structures.
Study on anti-de Sitter structures on surfaces with punctures.
problem Deformation of anti-de Sitter structures on surfaces with punctures.
method Parameterization of deformation space using Teichmüller spaces and quadratic differentials.
result Two parameterizations of the deformation space of wild globally hyperbolic anti-de Sitter structures.
Study describes how to realize periods of meromorphic differentials with specific properties.
problem Realizing meromorphic differentials with given zeros, poles, and topological constraints.
method Complete description of period representations for specified conditions on Riemann surfaces.
result A comprehensive method for realizing meromorphic differentials with prescribed characteristics.
In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved…
We introduce on any smooth oriented minimal surface in Euclidean 3-space a meromorphic quadratic differential, P, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of particular interest is the realization of its real part as a conservation law for a natur…
This paper classifies components of meromorphic differential strata.
problem Understanding the boundary of multi-scale compactification of meromorphic differentials.
method Classifying connected components of residueless meromorphic differentials.
result Classification of connected components of strata of residueless meromorphic differentials.
Classifies connected components of meromorphic differentials with residue conditions.
problem Understanding the structure of meromorphic differentials with residue constraints.
method Analyzes the multi-scale compactification and residue conditions.
result Classified connected components of generalized strata of meromorphic differentials.
Constructs a function to prove meromorphic differential strata don't have complete subvarieties.
problem Proving meromorphic differential strata don't contain complete subvarieties.
method Explicit construction of a strictly plurisubharmonic function.
result Proves meromorphic differential strata do not contain positive-dimensional complete subvarieties.
Let (Σ,p) be a pointed Riemann surface of genus g≥1. For any integer k≥1, we parametrize the space of meromorphic quadratic differentials on Σ with a pole of order (k+2) at p, having a connected critical graph and an induced metric composed of k Euclidean half-planes. The parameters form a finite-…
Study periodic orbits in specific parts of surface moduli spaces.
problem Understanding periodic orbits in the thin part of strata.
method Analyzing Teichmueller flow on specific subsets of moduli spaces.
result Asymptotic growth rate of periodic orbits in thin parts is at least h(Q)-1.
The paper studies the geometric structure of modular curves X1(N) using meromorphic differentials.
problem Understanding the topological complexity of modular curves X1(N). method Using the walls-and-chambers structure of strata of meromorphic differentials.
result Formulas for the number of chambers and effective means to draw the incidence graph of X1(N). Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.
problem Classifying Teichmüller curves in hyperelliptic components of meromorphic strata.
method Non-existence criterion based on intersections with moduli space boundary.
result Contradiction to algebraicity of candidate Teichmüller curves outside Hurwitz covers.
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…
Study meromorphic k-differentials with prescribed singularities on Riemann surfaces.
problem Understanding local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeros and poles, and k-residues at poles.
result For a given pattern of zeros, there exists a primitive holomorphic k-differential with these zeros.
Geometric approach to meromorphic differentials' periods and their holonomy representations.
problem Characterizing representations of meromorphic differentials' periods.
method Constructing translation structures with prescribed holonomy.
result Generalization of Haupt's classical result to meromorphic differentials.
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
Study linear subvarieties of meromorphic differential strata, proving toric closures and new proofs of theorems.
problem Understanding linear subvarieties in strata of meromorphic differentials.
method Investigate closures in multi-scale compactification, prove restrictions on period coordinates.
result Prove closures are locally toric varieties, generalize cylinder deformation theorem.
The study bounds the complexity of meromorphic differentials' directions.
problem Understanding the descriptive complexity of meromorphic differentials.
method Geometric lemma and topological analysis of saddle connections.
result Sharp upper bound on the Cantor-Bendixson rank of meromorphic differentials.
Classifies components of strata of k-differentials on Riemann surfaces.
problem Classifying connected components of strata of k-differentials.
method Developed new techniques to study connected components of strata of k-differentials for general k.
result Complete classification of connected components of the strata of quadratic differentials with arbitrary poles.
Complex nilmanifolds have limited algebraic dimension based on holomorphic differentials.
problem Determining the algebraic dimension of complex nilmanifolds.
method Analyzing holomorphic differentials and meromorphic maps on complex nilmanifolds.
result The algebraic dimension a(M) is bounded by the dimension of the space of holomorphic differentials. The paper defines and computes volumes of meromorphic differentials with simple poles.
problem Defining and computing volumes of strata of meromorphic differentials with simple poles.
method Definition of volume as an integral of a tautological class, computation by induction, and solution of an integrable system.
result Algebraic constants of volumes can be computed and shown to be solutions of integrable systems.
Computes cohomology classes of strata of meromorphic differentials.
problem Computing cohomology classes of differentials with prescribed poles.
method Defined a space of stable meromorphic differentials, stratified by zero multiplicities, computed Poincaré-dual cohomology classes, proved tautological, and provided an algorithm.
result All cohomology classes are tautological and can be computed.
We count meromorphic differentials with fixed residues and poles of fixed orders.
problem Counting meromorphic differentials with fixed residues and poles of fixed orders.
method Intersection theory on compactified moduli spaces of differentials.
result Complete solution to the problem with interesting combinatorial properties.