Study finds new factorable surfaces with non-zero curvature in pseudo-Galilean space.
problem Classifying surfaces with non-zero curvature in pseudo-Galilean space.
method Analyzing factorable surfaces as graphs of product functions.
result New classification results for factorable surfaces with non-zero Gaussian and mean curvature.
The paper classifies and finds surfaces with specific curvature in isotropic spaces.
problem Finding surfaces with prescribed Gaussian and mean curvature in isotropic spaces.
method Classified and found surfaces of different types with specific curvature conditions.
result Affine factorable surfaces of different types with prescribed curvature were found and classified.
Positive factorization for pseudoperiodic homeomorphisms on surfaces.
problem Factorization of pseudoperiodic homeomorphisms on surfaces.
method Generalization of classical results on smooth germs of surfaces, topological characterization of monodromies, and use of positive factorization criteria.
result Pseudoperiodic homeomorphisms on surfaces with positive fractional Dehn twist coefficients and screw numbers admit a positive factorization.
Study stretch factors on nonorientable surfaces, proving nonorientable techniques ineffective.
problem Determine stretch factors on nonorientable surfaces.
method Analyzing pseudo-Anosov maps with orientable foliations on specific surfaces.
result Stretch factors do not have Galois conjugates on the unit circle.
Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
The paper classifies surfaces with constant curvature in isotropic space.
problem Classifying surfaces with constant curvature in isotropic space.
method Classification based on constant Gaussian and mean curvature.
result Non-existence result for surfaces with H/K=const.
The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a d…
Minimal stretch factor for non-orientable surfaces is small.
problem Finding the minimal stretch factor for pseudo-Anosov homeomorphisms on non-orientable surfaces.
method Adapting Thurston's theory of fibered faces for non-orientable 3-manifolds.
result The minimal stretch factor is asymptotically on the order of 1/g.
Explains conformal maps using twistor lifts and provides a factorization method.
problem Understanding and analyzing conformal maps from a surface to Euclidean 4-space.
method Uses twistor lifts and differential factorization to explain and analyze conformal maps.
result Obtains a local factorization of the differential of a conformal map.
Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.
problem Finding stretch factors for weak Perron numbers.
method Constructing an end-periodic homeomorphism on a surface.
result Every weak Perron number is an end-periodic stretch factor.
Study monodromy factorizations for lines on del Pezzo surfaces.
problem Understanding monodromy factorizations for lines on del Pezzo surfaces.
method Listed monodromy factorizations in the mapping class group of a torus.
result Explicit correspondence between factorizations and roots of E8.
The abstract discusses braided surfaces and their characteristic maps, linking them to algebraic and geometric properties.
problem Characterizing braided surfaces and their characteristic maps.
method Using branched coverings and factorization through free groups, the abstract explores the algebraic and geometric properties of braided surfaces.
result The abstract establishes a connection between braided surfaces and characteristic maps, providing new insights into their structure.
The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface S can be as high as the dimension of the Teichmüller space of S. In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anoso…
We study diffeomorphisms of compact, oriented surfaces, developing methods of distinguishing those which have positive factorizations into Dehn twists from those which satisfy the weaker condition of right veering. We use these to construct open book decompositions of Stein-fillable 3-manifolds whose monodromies have n…
The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.
problem Understanding the factorization of harmonic maps between Riemann surfaces and manifolds.
method The proof relies on geometric properties of the Hopf differential and properties of holomorphic and anti-holomorphic diffeomorphisms.
result The theorem provides a factorization of harmonic maps under certain conditions involving holomorphic or anti-holomorphic diffeomorphisms.
Paper studies flow on hyperbolic surfaces to match boundary lengths.
problem Matching boundary lengths of hyperbolic surfaces.
method Combinatorial Yamabe flow on hyperbolic bordered surfaces.
result Flow converges exponentially to a surface with equal boundary lengths.
Unified representation for minimal and constant mean curvature surfaces.
problem Representing minimal and constant mean curvature surfaces in Euclidean and hyperbolic spaces.
method Integral system methods applied to Weierstrass and Bryant representations.
result Unified representation and classification of various examples.
Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.
problem Understanding the relationships between different types of surfaces and their integrable systems.
method Unified view through families of flat connections and parallel sections.
result Complete description of links between different surface types and their dressing transformations.
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…
Solves a Dirichlet problem for flat metrics on Riemann surfaces with boundary.
problem Solving a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary.
method Proves solvability using flat hermitian metrics and factorization results.
result Solves the Dirichlet problem for flat metrics on Riemann surfaces with boundary.
The paper classifies surfaces in isotropic spaces satisfying specific curvature relations.
problem Classifying surfaces in isotropic spaces with given curvature relations.
method Analyzing surfaces in isotropic 3-space I^3 with the relation aK+bH=c and K=H^2.
result Complete classification of linear Weingarten factorable surfaces and graph surfaces in I^3.
Maps between surfaces have degree constraints based on their Euler characteristics.
problem Constraints on the degree of maps between surfaces based on their Euler characteristics.
method Used the Kneser-Edmonds factorization theorem and provided a simple proof.
result Maps between surfaces have degree constraints based on their Euler characteristics.
The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.
problem Finding hyperbolic bordered surfaces with prescribed boundary lengths.
method Fractional combinatorial Calabi flow and generalized combinatorial Yamabe flow.
result The flows converge to a hyperbolic surface with prescribed boundary lengths.
New characterization of symplectic surfaces in CP^2 via bridge trisections.
problem Characterize symplectic surfaces in CP^2.
method Bridge trisections and quasipositive factorizations.
result Minimal genus symplectic surfaces are isotopic to surfaces in transverse bridge position.
We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N i…
The computation of the cobordism group of Morse functions on unoriented surfaces using Stein factorizations.
Minimal stretch factors for certain pseudo-Anosov maps are bounded.
problem Bounding the stretch factor of orientation-reversing pseudo-Anosov maps.
method Using the silver ratio and properties of puncture orbits to derive bounds.
result The bound λ(f)−χ(S)≥σ2 is asymptotically sharp. Study on special anisotropic conformal changes of conic pseudo-Finsler surfaces.
problem Exploring various anisotropic conformal transformations of conic pseudo-Finsler surfaces.
method Presented various anisotropic conformal transformations including C-anisotropic, horizontal C-anisotropic, and vertical C-anisotropic transformations. result Vertical φT-condition transformation makes every Landsberg surface Berwaldian. Study conical Ricci flow on surfaces with explicit asymptotic expansions.
problem Analyzing Ricci flow on conical surfaces with explicit regularity.
method Established framework for linear parabolic equations on conical surfaces; proved long-time existence and optimal regularity of conical Ricci flow.
result Explicit asymptotic expansions of conformal factor for conical Ricci flow.
Paper disproves a Farb question about pseudo-Anosov homeomorphisms.
problem Relation between pseudo-Anosov homeomorphisms and surface genus.
method Analyzes algebraic degree of stretch factor vs. surface genus.
result Negative answer to Farb's question about pseudo-Anosov homeomorphisms.
We study on a new kind of surface covered by translation and factorable (TF-type) surfaces in the three dimensional Euclidean space. We consider I and III Laplace-Beltrami operator surfaces of a TF-type surface. Then we obtain degrees and classes of algebraic surfaces of the surfaces using eliminate methods on software…
In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of Δn2 where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentatio…
A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a me…
We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of SL(2,R). We prove a fuchsian affine action of a surface group is never proper.
We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinit…
Paper solves injectivity of X-ray transform on surfaces.
problem Injectivity of non-Abelian X-ray transform on surfaces.
method Factorization theorem for Loop Groups, energy methods, scalar holomorphic integrating factors.
result Extends results to arbitrary Lie groups.
The paper shows how surjective homomorphisms between surface braid groups factor and computes their automorphism groups.
problem Characterizing surjective homomorphisms and automorphisms of surface braid groups.
method Analyzing the structure of surface braid groups and their homomorphisms.
result Surjective homomorphisms factor through forgetful maps and automorphisms are geometric.
The paper studies surface bundles and Dehn twists, providing new bounds and factorizations.
problem Understanding stable commutator lengths of Dehn twists and their gaps in mapping class groups.
method Examples of surface bundles, factorizations of Dehn twists, and asymptotic bounds.
result Improved upper bounds for stable commutator lengths and a gap in mapping class groups.
New symplectic surfaces and 4-manifolds created via genus-3 pencils.
problem Creating symplectic surfaces and 4-manifolds homeomorphic but not diffeomorphic.
method Explicit construction of symplectic genus-3 Lefschetz pencils.
result Infinite family of symplectic Calabi-Yau surfaces with b_1=2,3,4.
The paper calculates spectral determinants for two complex surfaces.
problem Calculating spectral determinants for complex surfaces.
method Closed explicit formulas, multiplicative relations, Belyi maps, and constant-curvature spheres.
result Spectral determinants of the Bolza surface and Klein quartic are calculated.
Solves open problem on simple surfaces with novel twistor correspondence.
problem Existence of nontrivial holomorphic vector bundles on simple surfaces.
method Novel twistor correspondence, Nash-Moser inverse function theorem, and microlocal analysis.
result Simple surface twistor space supports no nontrivial holomorphic vector bundles.
The paper bounds crossing numbers of dense graphs on surfaces.
problem Estimating the minimum number of edge crossings for dense graphs on surfaces.
method Proved lower and upper bounds on crossing numbers, providing explicit families of surfaces.
result Upper and lower bounds on crossing numbers match up to constant factors.
Skew parallelogram nets factorize, encompassing discrete differential geometry.
problem Factorization of polynomials in discrete differential geometry.
method Lax representation, Bäcklund transformations, factorization of polynomials.
result Skew parallelogram nets encompass all systems with polynomial representations.
New proof shows homomorphisms from pure braid groups to hyperbolic groups have cyclic images or factor through forgetful maps.
problem Characterizing homomorphisms from pure braid groups to hyperbolic groups.
method Extending and proving a new rigidity result for pure braid groups, focusing on homomorphisms to hyperbolic groups.
result Homomorphisms from pure braid groups to hyperbolic groups either have cyclic images or factor through a forgetful map.
Study harmonic maps with constant energy on curved surfaces.
problem Harmonic maps with equal energy on negatively curved surfaces.
method Analyzes complex submanifolds of Teichmüller space and uses factorization results.
result Shows existence of a closed Riemann surface Y such that harmonic maps factor through φX:XoY. A new ranking model uses nonnegative matrix factorization for tennis players.
problem Modeling latent variables influencing tennis player performance.
method Combines Bradley-Terry-Luce model with nonnegative matrix factorization.
result Model identifies surface type as key determinant of male player performance.
We explicitly construct pseudo-Anosov maps on the closed surface of genus g with orientable foliations whose stretch factor λ is a Salem number with algebraic degree 2g. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d s…
We prove a multiplicity formula for Riemann-Roch numbers of reductions of Hamiltonian actions of loop groups. This includes as a special case the factorization formula for the quantum dimension of the moduli space of flat connections over a Riemann surface.