The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Paper shows maps from m-sphere to n-sphere are trivial cobordant.
problem Understanding cobordism classes of maps and covers for spheres.
method Analyzes cobordism classes of maps from m-sphere to n-sphere.
result For m>n, cobordism classes of maps from m-sphere to n-sphere are trivial.
New method for triharmonic maps to spheres in various dimensions.
problem Creating triharmonic maps to spheres in different dimensions.
method Construction method based on eigenmaps and suitable deformations.
result Existence of triharmonic maps from Rm∖{0} into spheres. New energy identity found for biharmonic maps into spheres.
problem Establishing energy identity for biharmonic maps in supercritical dimensions.
method Adapting Lin-Rivière's strategy for sphere-valued maps.
result Energy identity for stationary biharmonic maps into spheres in supercritical dimensions n≥5. Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…
Standard special generic maps solve a homotopy sphere problem.
problem Determining integers p for which homotopy spheres admit special generic maps into Euclidean spaces.
method Using Stein factorization, we introduce standard special generic maps and study their properties.
result Standard special generic maps give a filtration of the group of homotopy spheres related to the Gromoll filtration.
Characterizes conformal Gauss maps into spheres.
problem Understanding conformal Gauss maps of surfaces.
method Invariant formulation and proof of characterisation of harmonic maps.
result Characterisation of harmonic maps as conformal Gauss maps of Willmore surfaces.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
We study biharmonic maps and f-biharmonic maps from a round sphere (S2,g0), the latter maps are equivalent to biharmonic maps from Riemann spheres (S2,f−1g0). We proved that for rotationally symmetric maps between rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to a 2nd order l…
Study harmonic maps into Hilbert-Schmidt Grassmannian.
problem Harmonic spheres conjecture.
method Twistor description of harmonic maps.
result Motivated by harmonic spheres conjecture.
Authors create stable proper biharmonic maps from unit ball to spheres.
problem Constructing stable proper biharmonic maps from compact domains.
method Established second variation formula of bienergy, examined stability of previously constructed maps.
result Existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.
In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sp…
The article explores constructing biharmonic and conformal biharmonic maps to spheres.
problem Constructing biharmonic and conformal biharmonic maps to spheres.
method Geometric algorithm to render harmonic maps biharmonic or conformally biharmonic.
result Explicit critical points for conformal-biharmonic maps between spheres are found.
Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree ±1. It would be interesting to know if there …
Existence and instability of biharmonic maps from balls to spheres.
problem Existence and stability of biharmonic maps between balls and spheres.
method Existence proof and instability analysis using bienergy.
result Existence of two proper biharmonic maps and instability in low dimensions.
Study on harmonic Gauss maps for submanifolds in Euclidean space and sphere.
problem Existence and non-existence of unit normal sections with harmonic associated Gauss maps.
method Proof of existence and non-existence results for submanifolds in Euclidean space and sphere.
result Obtained applications to CMC hypersurfaces of the sphere and isoparametric submanifolds.
Study invariants of Z/p-homology 3-spheres from abelianization of mapping class groups.
problem Deciding and constructing invariants of Z/p-homology 3-spheres. method Formulating a criterion and using families of trivial 2-cocycles on the abelianization of the level-p mapping class group. result Disproved a conjectured extension of the Casson invariant for rational homology 3-spheres.
In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes special generic maps on spheres and such maps have been constructed on smooth bundles …
Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps, derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese…
Classifies low energy maps from curved surfaces into spheres.
problem Classifying maps from surfaces of constant curvature into spheres.
method Analyzes maps with low energy and degree ±1, focusing on bubble configurations.
result Maps are quantitively close to a bubble configuration with specific radii.
Develops a framework for generating harmonic maps from a unit ball to a sphere.
problem Creating families of harmonic maps from a unit ball to a sphere.
method Based on Toth's machinery for generating eigenmaps, combined with a generalized radial projection.
result Provides theoretical background for constructing solutions to variational problems.
Study on harmonic maps from projective plane to 4D sphere, showing empty spaces for even degrees.
problem Investigating harmonic maps from projective plane to 4D sphere.
method Using twistor lifts to analyze spaces of harmonic maps.
result Spaces are empty for even harmonic degree and path-connected for harmonic degree less than 6.
The study finds infinitely many p-harmonic maps between spheres for specific p and m.
problem Investigating p-harmonic maps between spheres for different dimensions and p-values.
method Analyzing rotationally symmetric p-harmonic maps and their stability.
result Existence of infinitely many p-harmonic self-maps of spheres for given p and m.
Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is homotopic to a constant…
{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to construct explicit fold maps, which are often difficult. In this paper, we construct new…
The paper characterizes biharmonic maps between spheres using polynomial functions.
problem Characterizing biharmonic maps between spheres using polynomial functions.
method Proved a characterization formula and constructed biharmonic maps.
result Classification of all proper biharmonic quadratic forms from spheres.
Study on manifolds that map to lower dimensions with specific critical points.
problem Characterizing manifolds that map to Rn−1 with round fold maps. method Analyzing smooth n-dimensional closed manifolds with n≥4 and classifying round fold maps up to C∞ A--equivalence. result Determine which manifolds admit round fold maps into Rn−1 and classify these maps. Study pseudo-spherical submanifolds with 1-type Gauss map in pseudo-spheres.
problem Characterize and classify pseudo-Riemannian submanifolds with 1-type pseudo-spherical Gauss map.
method Classify Lorentzian surfaces and pseudo-Riemannian submanifolds in pseudo-spheres with 1-type pseudo-spherical Gauss map.
result Classification of submanifolds with 1-type pseudo-spherical Gauss map in pseudo-spheres.
Examines how meridians and parallels help in map drawing.
problem Understanding map drawing and foliations of the sphere.
method Analyzes Euler's work on cartography and meridians/parallels.
result Meridians and parallels are crucial for map drawing.
Classifies finite orbits of mapping class group action on character varieties.
problem Classifying finite orbits of mapping class group action on character varieties of punctured spheres.
method Inductive proof using Lisovyy--Tykhyy's classification for 4-punctured spheres as base case.
result Proves no finite orbits for 7-punctured spheres and unique 1-parameter family for 6-punctured spheres.
We construct homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere with arbitrarily small k-dilation for each k greater than (m + 1)/2. We prove that homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere cannot have arbitrarily small k-dilation for k less than or equal …
The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.
problem Whether rational homology n-spheres admit special generic maps into Rp for p<n. method Stein factorization technique to derive a necessary homological condition.
result New results on the (non-)existence of special generic maps for specific rational homology spheres.
Study improves Heegaard Floer homology relations for knots in homology spheres.
problem Improving relations in Heegaard Floer homology for knots in homology spheres.
method Proved inequality for d-invariants, used reduced Floer homology rank relations.
result Degree one maps between aspherical Seifert homology spheres are homotopic to homeomorphisms if Heegaard Floer homologies are isomorphic.
New invariant for CR maps from spheres discovered.
problem Identifying CR maps from spheres.
method Introducing a CR analogue of the Ahlfors derivative.
result The invariant distinguishes many sphere maps and vanishes for linear embeddings.
We construct the first known examples of nontrivial, normal, all pseudo-Anosov subgroups of mapping class groups of surfaces. Specifically, we construct such subgroups for the closed genus two surface and for the sphere with five or more punctures. Using the branched covering of the genus two surface over the sphere an…
Smooth maps show Gromoll filtration for spheres.
problem Understanding Gromoll filtration for special generic maps.
method Analyzing smooth maps with definite folds and Gromoll filtration.
result Gromoll filtration equals p for special generic maps.
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.
Study calculates virtually-cyclic dimension for specific mapping class groups.
problem Calculating virtually-cyclic dimension for mapping class groups of specific surfaces.
method Analyzes mapping class groups of punctured spheres with up to six punctures.
result Determines virtually-cyclic dimension for specific surfaces.
The paper proves a gap theorem for special harmonic maps between spheres.
problem Understanding the behavior of α-harmonic maps between spheres. method Analysis of approximations and energy identities to derive a gap theorem.
result An optimal gap theorem for α-harmonic maps of specific degrees. Classifies surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss maps.
problem Classifying surfaces in a pseudo-sphere with specific properties.
method Complete classification through Riemannian and Lorentzian surfaces, explicit systems of partial differential equations.
result Complete classification of surfaces with harmonic or 1-type pseudo-spherical Gauss maps.
Smooth approximations of Lipschitz maps via Ehresmann fibrations and Reeb sphere theorem for functions.
problem Approximating Lipschitz maps and understanding singular points in Riemannian manifolds.
method Using Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions.
result A Lipschitz map can be approximated by a smooth map via Ehresmann fibrations.
Constructs stable maps from 3-manifolds to surfaces without cusps.
problem Creating stable maps from 3-manifolds to surfaces without problematic points.
method Visual construction of stable maps with specific properties.
result Obtains stable maps with no cusps and specific fiber structures.
New maps show some surfaces can't be sections of 4D spheres.
problem Finding sections for certain 4D sphere maps.
method Exhibited singular fibrations with high genus fibers.
result Some regular fibers cannot be sections.
Study on group actions on spheres using multisymplectic geometry.
problem Existence of homotopy comoment maps for compact Lie group actions on spheres.
method Investigation of multisymplectic actions and comoments on spheres.
result Explicit constructions of comoments for interesting cases.
The paper constructs Willmore two-spheres using harmonic maps into a specific Lie algebra.
problem Constructing Willmore two-spheres via harmonic maps.
method Recalling isotropic harmonic maps and deriving their properties to describe Willmore two-spheres.
result Constructs examples of totally isotropic Willmore two-spheres and their adjoint transforms.
The paper constructs simplicial maps of any degree on spheres, solving a long-standing problem.
problem Constructing simplicial maps of any degree on spheres.
method Using connected sums and facet orientations, the paper develops a method to construct maps of any prescribed degree.
result The paper answers a question posed by Ryabichev and constructs simplicial maps of degree d for large d. This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
Study on sphere-valued maps, proving energy convergence and current limits.
problem Understanding the behavior of sphere-valued Sobolev maps as their energy grows.
method Proving Gamma-convergence of p-energies to the mass of an integral current. result Jacobian convergence to an area-minimizing current in a cobordism class.