Uniformizes surfaces with boundaries, focusing on triple junctions.
problem Uniformization of surfaces with boundaries, especially triple junctions.
method Extends conformal structure results to triple junction surfaces.
result Weak uniformization results for triple junction surfaces.
Paper proves discrete uniformizations converge to continuous for surfaces of genus ≥1.
problem Computing uniformizations for surfaces of genus >1.
method Discrete conformality and uniformization on triangle meshes.
result Discrete uniformizations approximate continuous uniformization for closed surfaces of genus ≥1.
New approach to nematic fields on surfaces, relaxing uniformity to quasi-uniformity.
problem Identifying least distorted nematic fields on generic surfaces.
method Relaxing the notion of uniformity into quasi-uniformity and proving parallel transport by geodesics.
result All quasi-uniform fields are parallel transported by the geodesics of the surface.
Survey on uniformization of metric surfaces, including fractal and topological manifolds.
problem Uniformization of metric surfaces homeomorphic to 2D topological manifolds.
method Various uniformization theorems, including quasisymmetric and quasiconformal approaches.
result Uniformization results for metric spheres and arbitrary metric surfaces.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
Paper generalizes discrete uniformization for genus-zero surfaces.
problem Discrete uniformization for surfaces of genus zero.
method Reduction to planar cases via stereographic projections.
result Generalization of discrete uniformization to genus-zero surfaces.
Uniform hyperbolicity proved for nonorientable surface curve graphs.
problem Proving uniform hyperbolicity for nonorientable surface curve graphs.
method Using bicorn curves and arguments from orientable surfaces.
result Graph of nonseparating curves is uniformly hyperbolic.
Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.
problem Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.
method Normalized Ricci flow on compact surfaces.
result Uniform Lipschitz continuity of isoperimetric profiles under normalized Ricci flow.
Simple Ricci flow proof for Riemann surfaces.
problem Uniformization theorem of Riemann surfaces
method Ricci flow and Hamilton's isoperimetric estimate
result Simple proof of uniformization theorem
Uniformizes surfaces using discrete harmonic maps and hyperbolic metrics.
problem Uniformizing surfaces with complex geometries.
method Least Dirichlet energy harmonic embedding of graphs on surfaces.
result Existence of hyperbolic metrics realizing least energy embeddings.
Uniform rectifiability proven for sets with Poincaré inequalities.
problem Uniform rectifiability of sets with Poincaré inequalities.
method Weak (1,d)-Poincaré inequality and surface measure. result Uniform rectifiability achieved for sets supporting such inequalities.
Study of Chern-Ricci flow on Hopf surfaces, showing finite-time volume collapse and uniform bounds.
problem Understanding the Chern-Ricci flow on Hopf surfaces, especially minimal non-Kähler ones.
method Construction of locally conformally Kähler metrics and analysis of Chern-Ricci flow.
result Finite-time volume collapse and uniform upper bounds on the metric tensor.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discr…
New approach finds minima of geodesic lengths for non-uniform fillings.
problem Finding minima of geodesic length functions for non-uniform fillings.
method Elementary optimization for 4-regular topological fillings, analysis of fat graphs and optimization techniques.
result Minima of geodesic length functions are found to be at triangle surfaces in both analyzed classes of non-uniform fillings.
Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.
problem Proving convergence of Chern-Ricci flow on complex minimal surfaces.
method Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence; surface torsion estimate, uniform total variation bound, Green-weighted L^2 estimate, linear iteration of real Poisson equations.
result Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence for normalized Chern-Ricci flow on complex minimal surfaces.
Study reveals uniform spectral gaps for random hyperbolic surfaces with few cusps.
problem Investigating spectral gaps for random hyperbolic surfaces with limited cusps.
method Analyzing Weil-Petersson random hyperbolic surfaces, showing no eigenvalues in specific intervals.
result Uniform lower bounds on spectral gaps for Weil-Petersson random hyperbolic surfaces, revealing a critical phenomenon of 'second order cancellation'.
In this note we clarify that the Rcci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.
The paper discusses methods to compute Green's function on algebraic surfaces using Schottky uniformization.
problem Computing Green's function on algebraic surfaces using Schottky uniformization.
method Investigates convergence of deformations of a formula related to Green's function.
result Provides insights into the geometric interpretation of the formula for Green's function.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
New approach to extremal hyperbolic surfaces using NEC groups.
problem Structural description of extremal hyperbolic surfaces.
method Uniformization by NEC groups for surfaces with cusps and/or geodesic boundary.
result Full description of automorphism groups of extremal surfaces.
Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures established.
problem Bounding the magnitude of harmonic Beltrami differentials and Weil-Petersson curvatures.
method Using the systole of a hyperbolic surface, the authors derive uniform bounds for the magnitude of harmonic Beltrami differentials and the Weil-Petersson Ricci curvature.
result Uniform bounds on Weil-Petersson curvatures and magnitudes of harmonic Beltrami differentials are established.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
New proof of uniformization for hyperbolic foliations.
problem Uniformization of foliated spaces by surfaces of hyperbolic type.
method Laminated Ricci flow to find a conformally equivalent metric with constant curvature -1.
result Existence of a laminated Riemannian metric with leaves of constant Gaussian curvature -1.
Uniformizes branched surfaces into Higgs bundles.
problem Uniformizing branched surfaces into cone metrics.
method Describes Higgs bundles corresponding to uniformization of conical metrics.
result Family of Higgs bundles parametrized by open subset of cohomology space.
We prove a uniform estimate, valid for every closed Riemann surface of genus at least two, that bounds the distance of any quadratic differential to the finite dimensional space of holomorphic quadratic differentials in terms of its antiholomorphic derivative.
The paper provides uniform length estimates for trajectories on flat cone surfaces.
problem Estimating the length of trajectories on flat cone surfaces.
method Using self-intersection numbers and constants depending only on the flat metric, the paper focuses on convex flat cone spheres with a positive curvature gap and a fixed number of singularities.
result Uniform two-sided estimates for trajectory lengths on convex flat cone spheres are obtained.
The paper proves uniform approximation for minimal surfaces with applications to a Mittag-Leffler theorem.
problem Approximating complete conformal minimal surfaces with finite curvature.
method Uniform approximation theorem with interpolation for minimal surfaces.
result Obtained a Mittag-Leffler type theorem for minimal immersions.
In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type mi…
Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.
problem Spectral gap for convex cocompact hyperbolic surfaces and their covers.
method Using thermodynamic formalism for twisted Selberg zeta functions.
result Uniform resonance-free regions for convex cocompact hyperbolic surfaces and expanders.
Classifies meromorphic affine connections on complex surfaces.
problem Investigating uniformization in higher dimensions with singularities.
method Extending work on holomorphic connections, classifying meromorphic connections on compact surfaces.
result Classification of meromorphic affine connections on compact complex surfaces.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense Hitchin representations.
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization the…
We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
problem Finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
method Discrete uniformization theorem, combinatorial α-Yamabe flow, combinatorial α-Calabi flow, edge flipping surgery.
result Longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery.
We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More gene…
The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.
problem The existence of thin surface subgroups in non-uniform arithmetic lattices.
method Analyzes arithmetic hyperbolic manifolds and their fundamental groups.
result Fundamental groups of non-compact arithmetic hyperbolic manifolds contain thin surface subgroups.
The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the ∂∂-class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces. result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the ∂∂-class of the Tricerri/Vaisman metric. Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.
problem Investigating uniform ellipticity and polyconvexity for anisotropic geometric energy functionals.
method Proves a variant of a recent result using real polyhedral chains.
result Uniform ellipticity of an anisotropic energy functional implies uniform polyconvexity of the integrand.
We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonsepara…
Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.
We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu−e2u=K0(z) on general open surfaces. A few oth…
The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characte…
We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform ε−regularity estimates whic…
In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic…
The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.
problem Building noncompact hyperbolic surfaces with uniform spectral gaps.
method Introduced a random graph model Fχ,n to construct expanding families of graphs, then applied these families to create hyperbolic surfaces. result Explicitly constructed an expanding family of graphs in the critical regime, leading to a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.
Harmonic maps from surfaces to CAT(k) spheres are branched coverings.
problem Uniformization of surfaces with CAT(k) metrics.
method Almost conformal harmonic maps and branched coverings.
result CAT(k) spheres are conformally equivalent to the 2-sphere.