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.
Study on Kodaira fibrations with nontrivial cohomology, proving properties of their structure.
problem Characterizing Kodaira fibrations with specific cohomology properties.
method Analyzing invariant rational cohomology and properties of holomorphic sections.
result Kodaira fibrations with invariant cohomology admit specific coverings and monodromies.
Classifies surfaces of section for Seifert fibrations.
problem Classifying surfaces of section for Seifert fibrations.
method Discussing branched coverings and relating surfaces of section to algebraic curves.
result Relates surfaces of section to algebraic curves in weighted complex projective planes.
Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.
problem Holomorphic projective structures and bundles on surfaces.
method Generalization of principal bundle of projective 2-frames to branched projective structures.
result Affine spaces of branched projective structures with given branching classes.
We study branched covering spaces in several contexts, proving that under suitable circumstances the cover satisfies the same upper curvature bounds as the base space. The first context is of a branched cover of an arbitrary metric space that satisfies Alexandrov's curvature condition CAT(k), over an arbitrary complete…
Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is studied as an example of bundle-like objects with discordant (sogo) transition functi…
Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.
problem Finding compatible harmonic metrics for rank 3 Higgs bundles in the Hitchin section.
method Defined a symmetric pairing and studied spectral curves as 2-sheeted branched coverings.
result Gave a condition for Higgs bundles on C or C∗ to have compatible harmonic metrics. We construct a branched center manifold in a neighborhood of a singular point of a 2-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of 2-dimensional curren…
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavio…
The paper constructs symplectic forms on 4-manifolds using branched coverings and holomorphic line bundles.
problem Constructing symplectic forms on 4-manifolds with rational symplectic forms.
method Using branched coverings and holomorphic line bundles, the paper constructs symplectic forms that are Kähler in a neighborhood of the 2-skeleton of the manifold.
result The paper proves the existence of a cohomologous symplectic form that is Kähler in a neighborhood of the 2-skeleton of the manifold.
Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of or…
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
The article proves properties of Seifert links and their cyclic branched covers.
problem Properties of Seifert links and their cyclic branched covers.
method Left-orderability, co-oriented taut foliations, and L-space properties. result Proves the ADE link conjecture for Seifert links. We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locall…
A map from 3-manifold skein to Lagrangian skein via holomorphic curve counting.
problem Counting holomorphic curves in cotangent bundles for 3-manifold skein.
method Skein-valued counting of holomorphic curves in branched covers.
result Wall-crossing formula for skein traces in branched covers.
We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifu…
For a given bundle ξ:E→M over a manifold, configuration-section spaces on ξ parametrise finite subsets z⊆M equipped with a section of ξ defined on M∖z, with prescribed "charge" in a neighbourhood of the points z. These spaces may be interpreted physically as spaces of fiel…
We give an overview of various recent results concerning the topology of symplectic 4-manifolds and singular plane curves, using branched covers and isotopy problems as a unifying theme. While this paper does not contain any new results, we hope that it can serve as an introduction to the subject, and will stimulate in…
The Teichmüller harmonic map flow deforms both a map from an oriented closed surface M into an arbitrary closed Riemannian manifold, and a constant curvature metric on M, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [1…
Paper analyzes origami slope gaps and their distribution, finding a unique pattern.
problem Analyzing slope gaps in origami surfaces.
method Derived slope gap distribution of a specific origami by considering return times under the horocycle flow.
result Found a unique distribution of origami slope gaps, not a sum of scaled Hall distributions.
New compact, negatively curved Einstein 4-manifolds found without being locally homogeneous.
problem Finding new compact, negatively curved Einstein manifolds of dimension 4.
method Construction via branched covers of hyperbolic 4-manifolds, interpolation, and perturbation.
result First examples of compact, negatively curved Einstein 4-manifolds that are not locally homogeneous.
Paper constructs counterexamples of higher solutions to self-duality equations.
problem Whether all real sections give rise to global solutions of self-duality equations.
method Using Willmore surfaces and generalized Whitham flow.
result Higher solutions exist on an open dense subset of the Riemann surface, not globally.
This is a survey paper on the topic of Weil-Petersson geometry of Teichmuller spaces. Even though historically the subject has been developed as a branch of complex analysis, the treatment here is from the view-point of differential geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and Wolpert ove…
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.
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…
Proper branched coverings are homeomorphisms on 3D balls or when branch set is empty.
problem Global injectivity of proper branched coverings on Euclidean balls.
method Analyzing the global injectivity of proper branched coverings defined on the Euclidean n-ball. result Proper branched coverings are homeomorphisms on 3D balls or when branch set is empty.
New criterion for branched covers between 2-spheres.
problem Existence of specific types of branched covers between 2-spheres.
method Combining complex analysis and topology techniques.
result Established a complete criterion for the existence of branched covers.
Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected close surface N with Euler's characteristic less than or equal to 0. This …
For surface-knots, branched covers with degree 3 have simplifying numbers <3.
problem Understanding numerical invariants of branched covering surface-knots.
method Analyzing numerical invariant (simplifying number) of branched covering surface-knots with degree 3.
result Branched covering surface-knots with degree 3 have simplifying numbers less than 3.
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.
Open and discrete maps with specific branch set images are equivalent to PL branched covers.
problem Understanding the equivalence of open and discrete maps and PL branched covers.
method Demonstrated that an open and discrete map f:SnoSn with a specific branch set image is equivalent to a PL branched cover up to homeomorphism. result Open and discrete maps with a specific branch set image are equivalent to PL branched covers.
Criteria found for knots not determined by their double branched covers.
problem Determining knots from their double branched covers.
method Criteria based on tunnel number one knots and their double branched covers.
result Provided criteria for knots not determined by their double branched covers.
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.
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.
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.
The paper studies which branched covers can be lifted to braided embeddings.
problem Which branched covers lift to braided embeddings?
method Analyzes conditions for liftability using Hansen's criterion and examples in different dimensions.
result Not all branched covers lift to braided embeddings; examples are provided in various dimensions.
New method finds unrealizable branched cover data.
problem Existence of branched covers between Riemann surfaces.
method Connecting spherical conic metrics to combinatorial constraints.
result New infinite sets of exceptional branching data found.
The paper details folding of branched covers of the 3-sphere over knots.
problem Understanding folding of branched covers of the 3-sphere over knots.
method Presenting detailed foldings of dihedral covers of the 3-sphere branched over torus knots.
result Detailed foldings of dihedral covers of the 3-sphere branched over torus knots are presented.
New method learns better branching policies for MILP problems.
problem Improving branch and bound search for solving MILP problems.
method Imitates strong branching rule with parameterized state of B&B search tree.
result Generalized policies outperform current state-of-the-art.
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.
Course on knots using branched coverings.
problem Determining knots using cyclic branched coverings.
method Cyclic branched coverings of knots.
result New insights into knot classification.
In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.
Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…
Formula compares metrics on branched coverings of line bundles.
problem Comparing metrics on branched coverings of line bundles.
method Formula to compare Quillen metrics on branched coverings.
result Formula for comparing Quillen metrics on branched coverings.
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.
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.
New examples show transverse knots are determined by their branched covers.
problem Transverse knots and their isotopy classes.
method Constructing and analyzing non-isotopic transverse knots with contactomorphic cyclic branched covers.
result Transverse isotopy classes of many transverse knots are determined by the contactomorphism type of their cyclic branched covers.
Finite distortion maps cannot have compact branch sets under growth conditions.
problem Understanding the structure of branch sets in mappings of finite distortion.
method Analyzing the asymptotic growth of distortion and constructing specific examples.
result The bound on the size of branch sets is strict and achievable.