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…
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48 results for “Riemann Surfaces”
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.
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 on smooth moduli space of Riemann surfaces with uniformization theorem.
problem Understanding the smooth moduli space of Riemann surfaces and their uniformization.
method Developed techniques of rational norm of homological marking and decomposition of probability measures.
result Closed Riemann surfaces are uniformizable by Schottky groups of Hausdorff dimension less than one.
Abstract: Generalizes Riemann's bilinear relations for surfaces of genus ≥ 2.
problem No new significant results, but highlights a known observation.
method Generalization of Riemann's bilinear relations.
result No significant new results obtained.
Harmonic metrics of generically regular semisimple Higgs bundles on non-compact Riemann surfacesmath.DG
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.
Mirzakhani studied Riemann surfaces and their spaces.
problem Understanding Riemann surfaces and their spaces.
method Survey of her 20 papers.
result Contributions to understanding Riemann surfaces and their spaces.
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.
A new method shows open Riemann surfaces have finite genus.
problem Determining the genus of open Riemann surfaces.
method Introducing a quotient space of forms to determine finite genus.
result Open Riemann surfaces have finite genus and can be embedded in a compact surface.
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.
The smallest eigenvalues of Riemann surfaces grow like 1/g^2 for large genus.
problem Understanding the behavior of small eigenvalues of Riemann surfaces as genus increases.
method Constructing specific Riemann surfaces with controlled boundary curve lengths and systole counts.
result The smallest eigenvalues of Riemann surfaces are uniformly comparable to 1/g^2 for large genus.
Let be a compact hyperbolic Riemann surface of genus . We call a systole a shortest simple closed geodesic in and denote by its length. Let be the maximal value that can attain among the compact Riemann surfaces of genus . 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.
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.
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.
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 -gonal actions.
Proves compactness for lamination spaces in complex manifolds.
problem Compactness of lamination spaces in complex manifolds.
method Introduced a Levy-Prokhorov metric topology on measured Riemann surface laminations and proved compactness using this topology.
result Compactness theorem for embedded measured Riemann surface laminations in almost complex manifolds.
SU(2) flat connections link to 3D geometry with cosmological constant.
problem Mapping flat connections on Riemann surfaces to 3D twisted geometry.
method Relating flat connection quantities to geometrical quantities in discrete 3D space.
result Moduli space of SU(2) flat connections generalizes phase space of twisted geometry.
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…
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.
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…
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…
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.
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 and isoperimetric inequality.
result Derived interior estimates, Harnack inequalities, and global 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 II: The scattering matrix and generalized period mappingsmath.DG
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.
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.
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.
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.
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.
Study estimates Bergman kernel on hyperbolic surfaces.
problem Estimating Bergman kernel on hyperbolic surfaces.
method Derive off-diagonal estimates for tensor-products of cotangent line bundles.
result Derived off-diagonal estimates for Bergman kernel.
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 .
result Proves recursion between volumes of moduli spaces of super hyperbolic surfaces using algebraic geometry.
Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on with two conical singularities.
The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.
problem Existence and properties of semi-arithmetic Riemann surfaces.
method Combining number theory and hyperbolic geometry to prove existence and properties of semi-arithmetic Riemann surfaces.
result Existence of infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.
We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus , and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action f…
Causal characters of zero mean curvature surfaces of Riemann type in the Lorentz-Minkowski 3-spacemath.DG
A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …