Extends harmonic maps compactification to punctured Riemann surfaces.
problem Compactifying Teichmüller spaces for punctured Riemann surfaces.
method Using harmonic maps rays to extend compactification.
result The compactification still coincides with Thurston's compactification.
Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.
problem Distribution of zeros of Gaussian sections on semipositive line bundles.
method Analysis of Bergman kernels and random zeros in high tensor powers.
result Equidistribution, large deviation estimates, central limit theorem, and number variances for zeros in the semi-classical limit.
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.
Let h be a complete metric of Gaussian curvature K0 on a punctured Riemann surface of genus g≥1 (or the sphere with at least three punctures). Given a smooth negative function K with K=K0 in neighbourhoods of the punctures we prove that there exists a metric conformal to h which attains this function…
Study of Fubini-Study forms on surfaces with punctures.
problem Analyzing Fubini-Study forms on surfaces with punctures.
method Using Hermitian metrics, holomorphic line bundles, and Kodaira maps.
result Fubini-Study forms grow polynomially near punctures.
Estimates Bergman kernel of punctured disk with improved results.
problem Estimating Bergman kernel of punctured disk.
method Techniques from \cite{SunSun} applied to punctured disk with Poincaré metric.
result Improved results on Bergman kernels of punctured Riemann surfaces near singularities.
Formula calculates index for CR operators on surfaces with boundary punctures.
problem Computing the index for Cauchy-Riemann operators on surfaces with boundary punctures.
method Large antilinear deformations method, generalized to punctured surfaces.
result Involves a non-standard weighted count of boundary zeros in the Euler characteristic term.
Study on infinite energy maps from surfaces to CAT(0) spaces.
problem Harmonic maps with infinite energy from Riemann surfaces to CAT(0) spaces.
method Estimates of energy growth near punctures, proof of uniqueness.
result Precise estimates of energy growth near punctures and proof of uniqueness of harmonic maps.
We show that the minimum of asymptotic translation lengths of all point-pushing pseudo-Anosov maps on any one punctured Riemann surface is one.
The study proposes a conjecture about the monodromy group of singular hyperbolic metrics and provides evidence and confirmations.
problem Understanding the monodromy group of singular hyperbolic metrics on Riemann surfaces.
method Using meromorphic differentials and affine connections, the study examines the monodromy group and confirms the conjecture for specific Riemann surfaces.
result The monodromy group of the singular hyperbolic metric is Zariski dense in PSL(2, R) and cannot be contained in certain Lie subgroups.
We consider a log-Riemann surface S with a finite number of ramification points and finitely generated fundamental group. The log-Riemann surface is equipped with a local holomorphic difffeomorphism $π: \mathcal{S} \to \C$. We prove that S is biholomorphic to a compact Riemann surface with finit…
Analyzes harmonic metrics for Higgs bundles over punctured surfaces.
problem Analyzing the moduli space of parabolic Higgs bundles with varying weights.
method Investigates the analytic dependence of harmonic metrics on weights and stable Higgs bundles.
result Shows the harmonic metric depends analytically on weights and stable Higgs bundles.
Abstract framework for two meromorphic forms on punctured surfaces.
problem Developing a framework for two meromorphic forms on punctured Riemann surfaces.
method Abstract framework with Teichmüller regularity, degeneration detection, and pushability.
result Existence of a surface carrying two meromorphic differentials realizing any prescribed restricted pair.
In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the p…
The paper extends Gelfand-Kapranov-Zelevinsky construction to hyperbolic Riemann surfaces with punctures.
problem Stratifying the space of weight vectors for hyperbolic Riemann surfaces with punctures.
method Analogous to Gelfand-Kapranov-Zelevinsky construction, associates polyhedral fans to hyperbolic Riemann surfaces with punctures.
result The secondary fan of a hyperbolic Riemann surface with punctures is the normal fan of a convex polyhedron, the secondary polyhedron.
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 study constructs new minimal surfaces with more ramified values than previously known.
problem Understanding minimal surfaces with finite total curvature and specific ramification properties.
method Systematic construction of meromorphic functions on punctured spheres.
result New minimal surfaces with νg=2.5 and Dg=1 on the four-punctured sphere. Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed with meromorphic cubic differentials whose poles are at the puncture…
In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps f…
New proof of harmonic map existence from punctured surfaces to crowned hyperbolic targets.
problem Existence of harmonic maps from punctured surfaces to specific hyperbolic targets.
method Using Teichmüller theory and Minsky's work on limiting harmonic maps, constructing the conformal limit and proving existence.
result Existence of harmonic maps from any punctured Riemann surface to a given crowned hyperbolic target.
We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoot…
For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on …
Constructs surfaces with constant mean curvature from Bessel equation.
problem Creating surfaces with constant mean curvature.
method From the Bessel equation, constructs immersions of the twice-punctured Riemann sphere into R^3.
result Family of constant mean curvature surfaces constructed.
New method to parametrize infinite Riemann surfaces with bounded triangulations.
problem Parametrizing infinite Riemann surfaces with bounded triangulations.
method Introducing bounded ideal triangulations and proving real-analyticity of the parametrization.
result Real-analytic parametrization of Teichmüller spaces for infinite surfaces with bounded triangulations.
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.
This paper has been withdrawn by the author
We prove that any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action. Our proof is based on the representation of the space of m-spin structures on a Riemann surface …
Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of d-bar operators on the Teichmuller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space M{g,n} in the sense of orbifolds where it can be written in terms of Mumford-M…
Infinite circle packings on surfaces with conical singularities are possible.
problem Finding hyperbolic metrics with prescribed angles and circle packings on surfaces with punctures.
method Using infinite triangulations and hyperbolic metrics, the approach involves identifying the underlying Riemann surface and ensuring the circle packing combinatorics match the given triangulation.
result There are infinitely many conical hyperbolic structures in a conformal class with a circle packing in the combinatorics of a given triangulation.
Survey of computations for Riemann surface moduli spaces, focusing on unstable homology.
problem Computing unstable homology of moduli spaces of Riemann surfaces.
method Integral, mod-2, and rational coefficient computations; use of homology operations.
result Explicit generators of unstable homology for most cases determined.
Reduces super Teichmüller space constraints for Ramond punctures.
problem Constraints on super Teichmüller space for Ramond punctures.
method Imposes constraints on odd coordinates of super Teichmüller space for monodromies around Ramond punctures.
result Reduces odd dimension to match moduli spaces of super Riemann surfaces.
The paper describes superconformal structures on super Riemann surfaces using fatgraphs.
problem Characterizing superconformal structures on super Riemann surfaces.
method Using fatgraphs to assign data, characterizing moduli and deformations with Strebel differentials and Čech cocycles.
result Superconformal structures on N=1 super Riemann surfaces are computed as fixed points of involution on N=2 super Riemann surfaces. Geometric proof of vanishing products of Chern classes in moduli space of rank 3 parabolic bundles.
problem Vanishing products of Chern classes in moduli space of rank 3 parabolic bundles.
method Using the Mehta-Seshadri correspondence and line bundles associated to torus bundles, proving vanishing via geometric sections.
result The ring generated by Chern classes vanishes below the dimension of the moduli space.
Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let P=Pg,k=rg,k×R+k be the space of parabolic structures over Riemann surfac…
We consider harmonic immersions in RN of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total Gauss curvature. The contribution of each end is a multiple of 2π, determined by…
We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting fe…
Compactifies Riemann surface metrics with conical singularities.
problem Constant curvature metrics on Riemann surfaces with conical singularities.
method Compactified spaces and study local deformation of metrics.
result Sharp regularity theorem for coalescing conic points.
Derives precise asymptotic expansion of Kähler-Einstein metric on punctured sphere.
problem Calculating the complete Kähler-Einstein metric on a punctured Riemann sphere.
method Uses Schwarzian derivative and modular forms to determine coefficients.
result Explicitly determines coefficients for 3 to 12 omitting points.
The main goal of this note is to show that the study of closed hyperbolic surfaces with maximum length systole is in fact the study of surfaces with maximum length homological systole. The same result is shown to be true for once-punctured surfaces, and is shown to fail for surfaces with a large number of cusps.
In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The…
In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic …
The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.
problem Compactifying character varieties of punctured surfaces.
method Projective compactifications using ideal triangulations and Komyo's method.
result Boundary divisors are toric varieties and the boundary complex is a sphere.
Study of sectorial decompositions in symmetric products of surfaces for symplectic geometry.
problem Symplectic topology of symmetric products of Riemann surfaces.
method Liouville sectorial techniques.
result New geometric proof of Homological Mirror Symmetry for a specific case.
Study character varieties of surfaces using cluster algebras and Poisson structures.
problem Character varieties of surfaces and their Poisson structures.
method Use Bonahon-Wong's trace map and cluster algebras associated with ideal triangulations.
result Recover Goldman Poisson algebra from cluster algebra structure and show automorphisms.
We prove that any two finite-area non-compact hyperbolic Riemann surfaces S and T have finite covers that are arbitrarily close in the normalized Weil-Petersson metric, where we normalize by dividing the square of the metric by the area of the surface. In the case where T is the modular surface this reduces to showing …
The paper studies complex affine structures near irregular singularities.
problem Understanding complex affine structures near irregular singularities.
method Introducing local invariants and a Delaunay decomposition.
result Upper bounds on the complexity of the Delaunay decomposition.
The study examines Teichmüller distances in lattices and punctured tori.
problem Distribution of Teichmüller distances in moduli spaces.
method Uniform distribution of lattices, calculation of Teichmüller distances.
result Identifies distribution of distances in Teichmüller space.
Proper superminimal surfaces in hyperbolic 4-space can be approximated by conformal immersions.
problem Finding conformal superminimal surfaces in hyperbolic 4-space.
method Analysis of holomorphic Legendrian curves in the twistor space of H4. result Proper conformal superminimal immersions can be approximated by smooth ones.