Study how convex structures on surfaces change.
problem Degeneration of convex structures on surfaces.
method Analyzing bordered surfaces for convex projective structures.
result Understanding how convex structures degenerate.
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48 results for “structured surfaces”
Two flat structures on minimal surfaces lead to a unique harmonic function.
problem Understanding flat structures on minimal surfaces.
method Illustration of two independent flat structures and their harmonic function.
result Uniqueness of Enneper's surface captured by a harmonic function.
This paper completes the classification of discrete conformal structures on surfaces.
problem Classifying discrete conformal structures on surfaces.
method Axiomatic approach and study of existing structures.
result Find new classes of discrete conformal structures, including generalized circle packing metrics.
This paper classifies discrete conformal structures on surfaces with boundary.
problem Classifying discrete conformal structures on surfaces with boundary.
method Axiomatic approach ensuring good geometric structure, classification based on triangulation and axioms.
result Unified and generalized existing discrete conformal structures on surfaces with boundary.
The paper explores properties of Pin structures on surfaces and their cobordism.
problem Understanding Pin structures on compact surfaces and their cobordism.
method Analyzes Pin structures on surfaces and their cobordism, providing a construction for dimensions up to two.
result Pin structures on surfaces differ by a diffeomorphism if and only if they are cobordant, but this does not extend to higher dimensions.
New Poisson structures defined on surface moduli spaces.
problem Generalizing Poisson brackets to quasi-surfaces.
method Defined quasi-Poisson brackets on quasi-surfaces.
result New Poisson structures on quasi-surfaces.
Stable generalized complex structures on certain surfaces are constant.
problem Existence of stable generalized complex structures on ruled surfaces.
method Analysis of sphere bundles over surfaces of genus ≥2.
result Stable generalized complex structures on these surfaces are of constant type.
Higher Spin Klein Surfacesmath.AG
We find all m-spin structures on Klein surfaces of genus larger than one. An m-spin structure on a Riemann surface P is a complex line bundle on P whose m-th tensor power is the cotangent bundle of P. A Klein surface can be described by a pair (P,tau), where P is a Riemann surface and tau is an anti-holomorphic involut…
Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.
problem Rigidity and existence of discrete conformal structures on surfaces with boundary.
method Axiomatic framework and classification of discrete conformal structures.
result Extends results by Guo-Luo and Guo to a general context.
Unique birational structure proven on Inoue surfaces.
problem Proving uniqueness of birational structures on Inoue surfaces.
method Generalizing a result by Bruno Klingler, proving uniqueness of structures.
result The natural -structure on an Inoue surface is the unique -structure.
We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that …
Introduces Möbius structures and hyperbolic ends for -surfaces in hyperbolic space.
problem Understanding -surfaces in hyperbolic space.
method Studies Möbius structures and hyperbolic ends.
result Applications to -surfaces in hyperbolic space.
Finite groups can be automorphism groups of translation surfaces with poles.
problem Existence of finite automorphism groups on translation surfaces with poles.
method Analyzing translation surfaces with poles and extending results to branched projective structures.
result Finite groups can be automorphism groups of translation surfaces with poles.
Study on K3 surfaces' collapsing and special Kähler structures.
problem Understanding the structure of K3 surfaces' collapsing metrics.
method Analyzing and establishing connections to SKSs and Jacobian elliptic K3 surfaces.
result Established a bijection between integral singular SKSs on and Jacobian elliptic K3 surfaces.
Study connects K3 surfaces to holomorphic metrics, solving complex structure variation.
problem Understanding complex structure variation on K3 surfaces.
method Using Picard-Fuchs equations and lattice polarizations.
result Explicit example of locally conformally flat holomorphic metric.
Segre quartic surfaces linked to minitwistor spaces with Einstein-Weyl structures.
problem Understanding the relationship between Segre quartic surfaces and minitwistor spaces.
method Using Penrose correspondence and detailed investigation of dual varieties.
result Determined the degrees and structure of components of dual varieties.
Maximal automorphisms found for complex projective structures.
problem Maximizing automorphisms in complex projective structures.
method Analyzing Fuchsian uniformizations and Galois Belyi curves.
result Fuchsian uniformizations of Hurwitz surfaces achieve maximal automorphisms.
Study on existence and structure of P-area surfaces in Heisenberg group.
problem Existence and structure of P-area minimizing surfaces in the Heisenberg group.
method Characterization of existence and structure using an underlying vector field N, proving existence even without satisfying boundary conditions, and applying Barrier condition.
result Existence of P-area minimizing surfaces under certain conditions, providing new understanding of the Heisenberg group.
Symplectic structure found on projective structures on surfaces with boundary.
problem Deformation space of projective structures on surfaces with boundary.
method Natural symplectic structure on the space, integrating the Adler-Gelfand-Dikii-space of the boundary.
result Space is a Hamiltonian space for the symplectic groupoid.
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Introduces zebra structures on surfaces for directional foliation.
problem Finding canonical representatives of homotopy classes on surfaces.
method Introduces zebra structures and uses triangulations with edge connections to prove existence.
result Canonical representatives exist for homotopy classes on surfaces with specific triangulations.
Study on surface geometry in Lie groups with CR structures.
problem Understanding surface curvature in Lie groups with CR structures.
method Defined Gauss and mean curvature in Tanaka-Webster geometry.
result Gave specific examples of surface curvature calculations.
We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.
Study on entropy of bulging deformations of projective structures on surfaces.
problem Entropy of bulging deformations in real projective structures on surfaces.
method Analysis of topological entropy in terms of bulging deformations.
result Construction of divergent structures with convergent topological entropy.
Proofs for decomposing branched affine surfaces into triangles and cylinders.
problem Decomposing branched affine surfaces into simpler geometric shapes.
method Proof of Veech's theorem and introduction of invariant α.
result Any pair of decompositions can be connected by flips.
Superintegrable systems on surfaces are classified geometrically.
problem Classifying superintegrable systems on conformal surfaces.
method Geometric structures on conformal surfaces, conformal covariant structural equations.
result Explicit set of algebraic equations defining superintegrable systems on all constant curvature surfaces.
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
Study of infinite type surfaces' mapping class groups via hyperbolic structures.
problem Understanding mapping class groups of infinite type surfaces.
method Definition of a topology on a moduli space of marked hyperbolic structures.
result Continuous action of mapping class groups on the marked moduli space.
New LP structures for punctured and unpunctured surfaces are discovered.
problem Understanding cluster algebras from surfaces with punctures and non-orientability.
method Extending quasi-cluster algebras to include punctures and adding laminations to surfaces.
result All punctured and unpunctured surfaces admit Laurent phenomenon structures.
We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary ob…
Superconvergence of differential structure for finite element methods on perturbed surface meshesmath.NA
Paper studies superconvergence on surface meshes using gradient recovery.
problem Proving superconvergence on deviated surfaces.
method Introduces geometric supercloseness and an algorithmic framework for gradient recovery.
result Validates theoretical results with numerical examples.
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
Study geometric quantization on K3 surfaces, showing spectral convergence.
problem Quantization of K3 surfaces from spectral perspective.
method Special Lagrangian fibrations and hyper-Kähler structures.
result Spectral convergence of -Laplacians on prequantum line bundles.
Grafting is a surgery on Riemann surfaces introduced by Thurston which connects hyperbolic geometry and the theory of projective structures on surfaces. We will discuss the space of projective structures in terms of the Thurston's geometric parametrization given by grafting. From this approach we will prove that on any…
The paper classifies degenerate almost complex surfaces in a nearly Kähler space.
problem Classifying degenerate almost complex surfaces in nearly Kähler spaces.
method Investigates two distinct cases based on the preservation of the tangent bundle under the almost product structure.
result Complete and explicit classification of degenerate almost complex surfaces in nearly Kähler spaces.
New rigidity theorem for Scherk's surfaces and flat structures.
problem Characterizing and proving uniqueness of minimal surfaces and flat structures.
method Combining curvature estimates and geometric harmonic functions to construct fresh uniqueness results.
result Periodic minimal surfaces admit new uniqueness results.
New complex manifolds found with flat structure.
problem Finding compact complex manifolds with flat affine structure.
method Using Lie groups with left-invariant complex structure.
result Retrieved Inoue surfaces in 2D.
Engel structures on complex surfaces are classified based on Chern classes.
problem Classifying complex surfaces with specific Engel structures.
method Using Chern classes and adapting Geiges' construction.
result Engel structures on complex surfaces with trivial Chern classes exist and are unique.
Study structural invariants of Goursat distributions related to curve singularities.
problem Understanding local invariants of Goursat distributions.
method Investigate structural invariants akin to curve singularities on surfaces.
result Relate structural invariants to small growth invariants in the sequel.
Generalized Calabi-Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are b…
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.
Study on deforming discrete conformal structures on surfaces with boundaries.
problem Deforming discrete conformal structures on surfaces with boundaries.
method Introduce combinatorial Ricci flow and combinatorial Calabi flow, establish longtime existence and global convergence of solutions.
result Effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
Study on K3 surfaces with collapsing metrics and nilpotent structures.
problem Exploring Ricci-flat metrics on K3 surfaces that collapse.
method Continuous map from K3 surface to an interval, with specific fiber types.
result Bubbles of Tian-Yau and Taub-NUT metrics appear as fibers.
The paper studies discrete centroaffine surfaces in 3D space.
problem Understanding centroaffine invariants and convexity of discrete surfaces.
method Developed structure equations and integrable systems for discrete centroaffine surfaces. Calculated centroaffine invariants and analyzed Laplacian operator.
result Obtained centroaffine invariants and studied their implications on surface convexity.
Classifies star log symplectic structures on surfaces.
problem Classifying star log symplectic structures on compact surfaces.
method Classifies bi-vectors with degeneracy loci modeled by lines intersecting at a point.
result Computes Poisson cohomology and discusses relationships with second cohomology.
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 super Riemann surfaces are computed as fixed points of involution on super Riemann surfaces.
Characterizes monodromies of projective structures on finite-type surfaces.
problem Understanding monodromies of projective structures on finite-type surfaces.
method Geometrical/topological study of local conical projective structures.
result Any representation can be represented as the holonomy of a branched projective structure.
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.