Local index theorem for cofinite hyperbolic Riemann surfaces derived from computational perspective.
problem Deriving the local index theorem for cofinite Riemann surfaces.
method Using Ahlfors' variational formulas and projection formulas, deriving integral formulas for variations of determinants.
result Explicit integral formulas for variations of logdetΔn and logdetNn. For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace sp…
We prove that the mapping class group Γg,n for surfaces of negative Euler characteristic has a cofinite universal space $\E$ for proper actions (the resulting quotient is a finite CW-complex). The approach is to construct a truncated Teichmueller space $\T_{g,n}(ε)$ by introducing a lower bound for the length of…
The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.
problem Understanding the structure of hyperbolic log-orbi curves.
method Formulates hyperbolic uniformization as a Tannakian reconstruction theorem and constructs a canonical maximal parahoric PSL2-Higgs object.
result Reconstructs the absolute Galois group of a one-variable complex function field as the inverse limit of etale fundamental groups of orbifold models.
Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a ne…
We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends.
The trace set of a Fuchsian group Γ ist the set of length of closed geodesics in the surface Γ\H. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithm…
For any positive integer n, we exhibit a cofinite subgroup Γn of the mapping class group of a surface of genus at most two such that Γn admits an epimorphism onto a free group of rank n. We conclude that H1(Γn;Z) has rank at least n and the dimension of the second bounded cohomology of each of these ma…
On the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form mπ for m=2,3,4,5,6math.GT We study arithmetic properties of the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form mπ for m=2,3,4,5,6 and show that the growth rates are always Perron numbers.
Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short intro…
A method to describe Riemann surfaces using graph profiles is proposed.
problem Describing Riemann surfaces in a graph-theoretic framework.
method Graph profiles to represent Riemann surfaces, establishing necessary and sufficient conditions for their existence.
result A criterion for the existence of Riemann surfaces with a given signature.
Topology on mapping class group leads to generic results on surface properties.
problem Defining and analyzing a topology on mapping class groups.
method Introduced a twist-cofinite topology on mapping class groups of surfaces.
result Generic properties of subsets, such as pseudo-Anosov maps and Heegaard manifolds.
Study Liouville action for harmonic maps between Riemann surfaces.
problem Optimizing harmonic maps between Riemann surfaces.
method Derive variational formula for Liouville action.
result Found variational formula for harmonic diffeomorphisms.
Every open Riemann surface can be triangulated with equilateral triangles.
problem The structure and triangulation of Riemann surfaces.
method Constructing a holomorphic branched covering to the Riemann sphere and glueing together equilateral triangles.
result Every open Riemann surface can be equilaterally triangulated.
In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi su…
The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of no…
Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.
problem Proving the existence and uniqueness of harmonic metrics on Higgs bundles.
method Analyzing Higgs bundles equipped with a non-degenerate symmetric pairing on non-compact Riemann surfaces.
result Proving the existence and uniqueness of compatible harmonic metrics under certain conditions.
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space Mg of closed Riemann surfaces, and is identified with the Schottky space of rank g. The main theorem of the paper is: Closed Riemann surfaces are uniformizable by S…
Criterion found for Teichmüller extremal maps on infinite Riemann surfaces.
problem Finding necessary and sufficient conditions for Teichmüller extremal maps.
method Established a criterion for Teichmüller-type extremal maps.
result Criterion for Teichmüller extremal maps on infinite Riemann surfaces.
New invariant for Riemann surfaces connects to moduli space classes.
problem Understanding Riemann surface invariants and their relationships.
method Introducing a twisted version of the Kawazumi-Zhang invariant and relating it to other classes.
result The new invariant connects to the first Mumford-Morita-Milller class on the moduli space.
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.
Let S be a compact hyperbolic Riemann surface of genus g≥2. We call a systole a shortest simple closed geodesic in S and denote by sys(S) its length. Let msys(g) be the maximal value that sys(⋅) can attain among the compact Riemann surfaces of genus g. We call a (global…
The paper discovers new ways Riemann surfaces can degenerate.
problem Understanding degeneration of infinite-type Riemann surfaces.
method Constructing a concrete example to prove degeneration phenomena.
result Existence of degenerations in Bers boundary of infinite-type surfaces.
New formula and algorithm for computing distances on complex Riemann surfaces.
problem Computing distances on higher-genus Riemann surfaces is challenging due to infinite terms in the formula.
method Derived a computable distance formula and developed an efficient algorithm.
result Reduced distance computation from an infimum to a minimum over a finite set of terms.
Estimates curvature for holomorphic maps on Riemann surfaces.
problem Curvature estimation for holomorphic maps on open Riemann surfaces.
method Use of jet differentials to establish a Gauss curvature estimate.
result Established a Gauss curvature estimate for holomorphic maps.
Algorithm constructs algebraic curves from translation surfaces.
problem Creating algebraic curves from translation surfaces.
method Using discrete Riemann surface theory and Riemann theta functions.
result Algorithm approximates Jacobian varieties of translation surfaces.
Proves open Riemann surfaces can be embedded into 4D space.
problem Embedding open Riemann surfaces in lower dimensions.
method Proper harmonic embedding by harmonic functions.
result Reduces embedding dimension from previously known 5D to 4D.
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.
Exact diameter found for some Riemann surfaces.
problem Calculating the diameter of compact Riemann surfaces exactly.
method Proved for a specific class of surfaces (generalized Bolza surfaces).
result Diameters of generalized Bolza surfaces are equal to their fundamental polygon radii.
Study the boundary of Riemann surfaces with abelian automorphisms.
problem Characterize the boundary of Riemann surfaces with abelian automorphisms.
method Analyze the moduli space and its Deligne-Mumford compactification, focusing on equisymmetric loci.
result Describe the topological strata at the boundary for hyperelliptic and cyclic p-gonal actions. Criterion for Lie algebroid connections on compact Riemann surfaces.
problem Finding conditions for Lie algebroid connections on compact Riemann surfaces.
method Analyzing stable holomorphic vector bundles and their connections.
result Necessary and sufficient condition for Lie algebroid connections on compact Riemann surfaces.
We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt super-Riemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition funct…
Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells havin…
New proof shows special surfaces have finite type.
problem Characterizing surfaces with finite topological type.
method Shorter proof of Huber's theorem.
result Special surfaces have finite topological type.
We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…
Study interior estimates for solutions of Poisson equation on Riemann surfaces.
problem Interior estimates for solutions of linear Poisson equation on Riemann surfaces.
method Used Zygmund space LlnL and isoperimetric inequality. result Derived interior estimates, Harnack inequalities, and global estimate.
Sharp estimate shown to be rigid on curved surfaces.
problem Sharp Bezout estimate on nonnegatively curved Riemann surfaces.
method General three circle theorem applied.
result Rigidity of the sharp Bezout estimate.
This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.
problem Constructing a discrete theory of real Riemann surfaces.
method Using quad-graphs and linear discretization of Cauchy-Riemann equations, constructing a symplectic homology basis.
result The discrete period matrix has the same canonical decomposition as in the smooth setting.
Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.
problem Scattering theory for harmonic one-forms on Riemann surfaces.
method Construction of scattering theory from boundary value problems involving systems of curves and jump problems. Explicit expression for scattering matrix using Schiffer operators.
result Unitary scattering matrix and general association of polarizing Lagrangian spaces.
This article is an attempt to generalize Riemann's bilinear relations on compact Riemann surface of genus at least 2, which may lead to new structures in the theory of hyperbolic Riemann surfaces. No significant result is obtained, the article serves to bring the readers' attention to the observation made by [Bol-1949]…
Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.
problem Proving a generalized Gauss-Bonnet formula for conical metrics.
method Analyzing Gaussian curvature and Lebesgue integrability.
result Proved a generalized Gauss-Bonnet formula for conical metrics.
Optimizes the first eigenvalues of Riemann surfaces for large genus.
problem Finding optimal lower bounds for first eigenvalues of Riemann surfaces.
method Analyzing shortest multi-closed curves to establish a new lower bound.
result The first eigenvalue of a Riemann surface is greater than a specific formula involving the genus and a constant.
Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.
problem Constructing moduli spaces over Riemann surfaces.
method Finite-dimensional construction using holomorphic symplectic reduction.
result Moduli spaces over Riemann surfaces as stratified holomorphic symplectic spaces.
SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Ri…
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
The paper studies geodesics on an icosahedron's Riemann surface.
problem Geodesics on a icosahedron's Riemann surface.
method Constructs a Riemann surface and analyzes geodesics on it.
result Discovers properties of geodesics under a flat metric.
Paper connects volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
problem Relating volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
method Relates volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces Mg,n. result Proves recursion between volumes of moduli spaces of super hyperbolic surfaces using algebraic geometry.