Discrete theory for rational maps improved by generalized branch points.
problem Discretization effects in locating branch points in circle packings.
method Introducing generalized branch points that can be positioned anywhere in the geometry.
result Fixed flaws in discrete Ahlfors and Weierstrasse functions using generalized branching.
New results on hypersurfaces show no branch points, improving smoothness.
problem Analyzing area minimising hypersurfaces mod p without branch points.
method General analysis of immersed stable minimal hypersurfaces with alternating orientation.
result Area minimising hypersurfaces mod p do not admit immersed branch points.
We show that, if the local dimension of the branch set of a discrete and open mapping f:M→N between n-manifolds is less than (n−2) at a point y of the image of the branch set fBf, then the local monodromy of f at y is perfect. In particular, for generalized branched covers between n-manifolds …
The study investigates how branch points affect the shape and mechanics of hyperbolic surfaces.
problem Understanding the role of branch points in the shape and mechanics of hyperbolic surfaces.
method Developed a discrete differential geometric (DDG) approach to study deformations of hyperbolic objects with distributed branch points.
result Branch points influence the overall morphology of hyperbolic surfaces without concentrating energy, leading to sub-exponential growth in maximum curvature.
Study continuous deformations of branched projective structures on surfaces, preserving holonomy and branch points.
problem Continuous deformations of branched projective structures on closed surfaces of genus g≥2. method Schiffer variations and analysis of canonical divisors.
result Branch points are necessarily arranged on a canonical divisor when the underlying complex structure is infinitesimally preserved.
Paper finds new realizable data for maps with three branch points.
problem Existence of rational maps with specific branch points.
method New families of branch data identified through football decomposition method.
result Identifies new realizable branch data and exceptional data.
Branch points of a real 2-surface S in a 4-manifold M generalize the branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when S is topologically embedded in M. It defines a link L(p), the components of which are clos…
The study examines the regularity of branched immersions using special coordinate systems.
problem Understanding the regularity of branched immersions and their fundamental elements.
method Development and use of special coordinate systems to express maps with branch points, proving existence and regularity conditions for mean curvature vectors.
result Characterization and existence of special coordinate systems for branch immersions, proving regularity conditions for mean curvature vectors.
We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projecti…
We desingularize a branch point p of a minimal disk F0(D) in R4 through immersions Ft's which have only transverse double points and are branched covers of the plane tangent to F0(D) at p. If F0 is a topological embedding and thus defines a knot in a sphere/cylinder around …
Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rac…
Uniformly branching trees are equivalent to certain metric spaces.
problem Characterizing metric spaces equivalent to uniformly branching trees.
method Proving equivalence between trivalent quasiconformal trees and uniformly branching trees.
result Any two uniformly branching trees are quasisymmetrically equivalent.
Researchers compute specific Hurwitz numbers for branched covers.
problem Computing the number of equivalence classes of surface branched covers.
method Combinatorial method based on Gronthendieck's dessins d'enfant.
result Explicit arithmetic formulae for weak Hurwitz numbers are derived.
The paper explores rational functions with 3 branching points on the Riemann sphere.
problem Existence of rational functions with specific branching points.
method Utilizes complex analysis to establish properties of rational functions.
result Identifies new types of exceptional branching data.
Ideal triangulations of 3-manifolds are shown equivalent up to certain moves.
problem Equivalence of ideal triangulations in 3-manifolds.
method Using branched triangulations and transit equivalences, the paper shows that ideal triangulations are equivalent up to certain moves.
result Ideal triangulations of 3-manifolds are equivalent up to certain moves.
Study on moduli spaces of branched projective structures on surfaces.
problem Characterizing and understanding moduli spaces of branched projective structures.
method Analytic and geometric methods to study the moduli spaces of branched projective structures.
result The moduli space of marked branched projective structures is a complex analytic space with specific dimensions and singular points.
In this paper we study the homeomorphisms of the disk that are liftable with respect to a simple branched covering. Since any such homeomorphism maps the branch set of the covering onto itself and liftability is invariant up to isotopy fixing the branch set, we are dealing in fact with liftable braids. We prove that th…
We deform a minimal disk in R4 with a branch point into symplectic minimally immersed disks with only transverse double points.
Simplifies surface-knots using chart moves involving black vertices.
problem Simplifying branched covering surface-knots.
method Chart moves involving black vertices to simplify surface-knots.
result Properties of simplified branched covering surface-knots with branch points.
Analyzes branch points of area-minimizing currents with non-2 planar frequency.
problem Understanding the structure of area-minimizing currents near branch points.
method Intrinsic frequency function and geometric arguments avoiding center manifolds.
result Establishes higher order asymptotics and topological control near branch points.
For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary re…
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.
Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.
problem Understanding complex wrinkling patterns in thin elastic hyperbolic surfaces.
method Non-Euclidean plate theory and investigation of branch points.
result Branch points are natural defects in hyperbolic sheets, influencing their morphology robustly.
Researchers create a partial resolution of Coulomb branches for gauge theories.
problem Understanding partial resolutions of Coulomb branches in gauge theories.
method Constructing partial resolutions as variants of generalized slices in geometric contexts.
result Identified partial resolutions with specific geometric objects.
Study of geodesic branching in 2D sub-Riemannian manifolds.
problem Branching of geodesics in sub-Riemannian manifolds of rank two.
method Analysis of geodesic behavior in sub-Riemannian geometry.
result Continuous families of strictly abnormal branching geodesics and accumulation of branching points.
Researchers compute specific types of branched surface covers.
problem Counting specific types of branched surface covers.
method Combinatorial method based on Grothendieck's dessins d'enfant.
result Explicit formulae for the number of covers in terms of local degrees.
The paper explores positivity and irreducibility in Hurwitz spaces related to differentials of the second kind.
problem Positivity and irreducibility in Hurwitz spaces of certain covers of the projective line.
method Analyzes strata of differentials of the second kind with fixed multiplicities of zeros and poles, and applies this to show positivity and irreducibility in Hurwitz spaces.
result The Hurwitz spaces of degree d, genus g covers of P1 with pure branching at all but possibly one branch point are irreducible under certain conditions. Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface ha…
Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane w…
Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limi…
Researchers study simple branched coverings and their cobordism groups.
problem Understanding cobordism groups of simple branched coverings.
method Constructing a universal k-fold simple branched covering and computing the module rationally.
result Determine the rank of cobordism groups and compute specific groups.
We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the branched surface). The first three conditions are…
Closed geodesic nets on surfaces have limited branch points
problem Geodesic nets on surfaces
method Bounding the number of branch points
result Proving a bound on branch points for closed geodesic nets
We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted version of the toroidal grid diagrams recently introduced by Manol…
Elementary proof shows no specific torus to sphere cover with certain branching points.
problem Existence of specific branched covers between torus and sphere.
method Elementary topological proof using properties of the torus.
result No such branched cover exists with specified branching points.
The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.
problem Analyzing harmonic maps between surfaces in the homotopy class of a covering map.
method Proving the uniqueness of critical points and injectivity of Hopf differential for harmonic maps.
result The uniqueness of critical points of energy function and injectivity of Hopf differential are proven under specific conditions.
We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of CP3, branched over the singular Kummer surface, with the branch locus in R(S2,6) corresponding to the binary dihedral representations. This follows from an analysis of the map induced on SU(2) c…
The paper extends Gauss-Bonnet and Hopf-Poincaré theorems to branched sections of fiber bundles.
problem Extending classical theorems to branched sections of fiber bundles.
method Defining index of singularity points, calculating examples, and proving Hopf-Poincaré-Gauss-Bonnet theorem for resolvable branched sections.
result Analog of Hopf-Poincaré-Gauss-Bonnet theorem for resolvable branched sections.
Classifies branched Willmore spheres using conformal Gauss maps.
problem Classifying branched Willmore spheres.
method Analyzing the asymptotic expansion of the conformal Gauss map at branched points.
result Full classification of branched Willmore spheres.
Generalized Thurston's characterization for branched coverings of the 2-sphere.
problem Characterize branched coverings of the 2-sphere.
method Introduced local balance and operations against balanced graphs.
result New proof of a theorem by Eremenko-Gabrielov-Mukhin-Tarasov-Varchenko.
Computes the monodromy of cubic surfaces branching over smooth cubic curves.
problem Understanding the monodromy of cubic surfaces.
method Computational approach using the relationship between inflection points and lines on cubic surfaces.
result The monodromy map is surjective onto the centralizer of the image of a generator of the deck group.
Study geodesics in sub-Riemannian manifolds, resolving open questions.
problem Understanding geodesics in sub-Riemannian geometry, especially those that lose regularity.
method Constructing examples and using a lifting procedure.
result Existence of non-smooth and branching minimizing geodesics in real-analytic sub-Riemannian manifolds and Carnot groups.
The paper studies triangulations of surfaces up to branched transit equivalences, proving equivalence conditions and foliations.
problem Understanding triangulations of surfaces up to branched transit equivalences.
method Analyzes triangulations of closed surfaces S with vertices V, considering branched triangulations up to b-transit equivalence generated by b-flips.
result Branched triangulations are equivalent under certain conditions, including parity of Euler-Poincare' characteristic c(S).
Rigidity of critical eigensections on spheres proven.
problem Rigidity of critical eigensections on spheres.
method Proved rigidity of critical eigensections through SO(3)-rotations.
result Minimal non-degenerate critical eigensections are deformation rigid.
We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critic…
Minimal surfaces in a Riemannian manifold Mn are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane RP2. We show that a minimal surface f:RP2→M3 which has the smallest area, among those ma…
For the existence of a branched covering Sigma~ --> Sigma between closed surfaces there are easy necessary conditions in terms of chi(Sigma~), chi(Sigma), orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also suf…
Machine learning finds Z/2 eigenfunctions on a sphere.
problem Finding Z/2 eigenfunctions on the sphere.
method Created a multivalued neural network and used JAX to implement it. Fixed branch points at tetrahedron and cube vertices, and allowed AI to move them in the third case.
result Found Z/2 eigenfunctions for three cases.