Compactness theorem for quasiregular maps between manifolds.
problem Compactness of sequences of quasiregular mappings.
method Gromov's compactness theorem for pseudoholomorphic curves.
result Subsequence of quasiregular mappings converges to a quasiregular map on a nodal manifold.
Notes on quasiregular maps between Riemannian manifolds, preserving Sobolev forms.
problem Extending quasiregular map theory from Euclidean to Riemannian manifolds.
method Recalling different approaches to first-order Sobolev spaces, showing equivalence, and transferring key theorems.
result Pull-backs with quasiregular maps preserve Sobolev differential forms of the conformal exponent.
Quasiregular maps get harmonic extensions in hyperbolic space.
problem Existence of harmonic extensions for quasiregular maps.
method Proving harmonic extensions for non-constant quasiregular maps on spheres.
result Non-constant quasiregular maps on spheres have harmonic extensions in hyperbolic space.
The paper proves a quasiregular cobordism theorem with applications in mapping theory.
problem Proving a quasiregular cobordism theorem for PL manifolds.
method Dimension-free deformation method for cubical Alexander maps.
result Existence of quasiregular maps with specific properties.
Study automorphic and Lattès-type maps on manifolds, finding restrictions on their groups.
problem Characterize automorphic and Lattès-type quasiregular maps on manifolds.
method Analyzing automorphic and Lattès-type maps under discrete groups of Euclidean isometries, considering multiplicity and group properties.
result Restrictions on the dimension of the group for automorphic and Lattès-type maps, especially in the Lattès case.
Extends pseudoholomorphic curves to quasiregular mappings between manifolds.
problem Generalizing pseudoholomorphic curves to mappings between Riemannian manifolds.
method Defines quasiregular curves as mappings satisfying a specific distortion inequality and proves their properties.
result Quasiregular curves satisfy Gromov's quasiminimality condition and Liouville's theorem.
Equivalence of quasiregular mappings on subRiemannian manifolds proven.
problem Equivalence of quasiregular mappings on subRiemannian manifolds.
method New Popp extensions of horizontal metrics and distortion estimates.
result All quasiregular mappings are equivalent with precise dependencies.
The study extends removability results for quasiregular curves in Euclidean spaces.
problem Removability of singularities in quasiregular curves.
method Extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves.
result Every non-constant quasiregular curve has infinite energy.
The paper constructs a multi-valued inverse of quasiregular maps and develops pull-back theory for differential forms.
problem Understanding multi-valued inverses of quasiregular maps and their properties.
method Using Almgren's framework of multi-valued maps and developing pull-back theory for differential forms.
result The multi-valued inverse is a quasiregular ω-curve with respect to a natural n-form ω. Quasiregular curves in product manifolds are shown to be carried by quasiregular maps.
problem Characterizing quasiregular curves in product manifolds with small distortion.
method Analyzing K-quasiregular volNimes-curves in product manifolds N=N1imes⋯imesNk. result Quasiregular curves of small distortion in product manifolds are carried by quasiregular maps.
Defines signed quasiregular curves and proves growth theorem.
problem Understanding growth of signed quasiregular curves.
method Proves weak reverse Hölder inequality and uses it to prove growth theorem.
result Proves growth theorem for signed quasiregular curves.
Generalizes cohomological obstruction for quasiregular ellipticity.
problem Cohomological obstruction for quasiregular ellipticity.
method Analyzes maps in Wloc1,n(Rn,M) with specific conditions. result Embeds real singular cohomology ring into exterior algebra in a graded manner.
The study establishes conditions for quasiregular mappings from Heisenberg groups to link complements.
problem Conditions for nonconstant quasiregular mappings from Heisenberg groups to link complements.
method Translation of a growth condition on fundamental groups into the existence of a supersolution to the 4-harmonic equation.
result A link complement admits a nonconstant quasiregular mapping from the Heisenberg group only if the link is empty, an unknot, or a Hopf link.
New mappings connect 3-sphere geometry to complex analysis problems.
problem Locally homeomorphic quasiregular mappings in 3-space and their properties.
method Constructing non-trivial cobordisms and equivariant mappings.
result Found new mappings related to complex analysis and geometry.
New classification for certain 4-manifolds using quasiregular mappings.
problem Classifying closed simply connected quasiregularly elliptic 4-manifolds.
method Using quasiregular mappings and de Rham cohomology.
result Closed simply connected quasiregularly elliptic 4-manifolds are homeomorphic.
New mappings solve long-standing problems in 3-space.
problem Longstanding problems for bounded locally homeomorphic quasiregular mappings in 3-space.
method Constructed bounded locally homeomorphic quasiregular mappings in the unit 3-ball using non-trivial hyperbolic cobordisms.
result Solved long-standing problems for quasiregular mappings in 3-space.
We discuss the issue of branching in quasiregular mapping, and in particular the relation between branching and the problem of finding geometric parametrizations for topological manifolds. Other recent progress and open problems of a more function theoretic nature are also presented.
We study mappings on sub-Riemannian manifolds which are quasi-regular with respect to the Carnot-Caratheodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit …
Compactness theorem for quasiregular curves proves normality and resolves nodal points.
problem Proving compactness and normality for quasiregular curves.
method Using Gromov's compactness theorem and bubble trees to associate a limit curve and measure.
result A nodal resolution of quasiregular curves via bubble trees.
We prove that the distortion function of the Gauss map of a harmonic surface coincides with the distortion function of the surface. Consequently, Gauss map of a harmonic surface is K quasiregular if and only if the surface is K quasiregular, provided that the Gauss map is regular or what is …
Local constancy of index for certain gradient mappings proved.
problem Proving the local constancy of the index for specific gradient mappings.
method Using a more general theorem for quasiregular gradient mappings, deducing the result from the Hessian's properties.
result The index is locally constant for C1,1 functions with uniformly positive determinant Hessian almost everywhere. Sharp bound on cohomology of quasiregularly elliptic manifolds.
problem Bounding cohomological dimension of manifolds with quasiregular mappings.
method Using combinatorial and geometric properties of cohomology, proving a sharp upper bound.
result A sharp upper bound on the cohomological dimension of manifolds, answering Gromov's question.
A map between manifolds is an isometry if it's Lipschitz and scalar curvature bounded.
problem Characterizing maps between manifolds based on their scalar curvature and Lipschitz continuity.
method Spectral properties of Dirac operators and index theory for low regularity metrics and bundles.
result A 1-Lipschitz map between manifolds is an isometry if it has bounded scalar curvature.
Improved Sobolev mappings in Carnot groups with weaker assumptions.
problem Improving Sobolev mappings in Carnot groups with weaker conditions.
method Using Buser-Karcher center-of-mass and polynomial expressions in moments.
result Rigidity and structural results hold under weaker Sobolev exponents.
Quasiregular curves are Hölder continuous and have higher integrability.
problem Understanding the Hölder continuity and integrability of quasiregular curves.
method Analyzing the Hölder continuity and integrability of curves defined by a K-quasiregular function with respect to a covector ω. result Quasiregular curves are (1/K)(∥ωVert/∣ω∣ℓ1)-Hölder continuous and have higher integrability. This article is the introductory part of authors PhD thesis. The article presents a new coordinate invariant definition of quasiregular and quasiconformal mappings on Riemannian manifolds that generalizes the definition of quasiregular mappings on Rn. The new definition arises naturally from the inner product struc…
First constructed genus 2 Cantor set in 3D space.
problem Constructing a geometrically self-similar Cantor set of genus 2.
method Geometrically self-similar construction in R3. result First uniformly quasiregular mapping with a genus 2 Cantor set Julia set.
The abstract manifold cannot have uniformly quasiregular self-maps.
problem Characterizing uniformly quasiregularly elliptic manifolds.
method Introducing conformally formal manifolds and proving their properties.
result The abstract manifold is not uniformly quasiregularly elliptic.
We study the existence of geometrically controlled branched covering maps from R3 to open 3-manifolds or to decomposition spaces S3/G, and from S3/G to S3.
The paper proves a rescaling principle for quasiregular curves and applies it to hyperbolicity.
problem Proving hyperbolicity for quasiregular curves.
method Rescaling principle for quasiregular curves into calibrated manifolds.
result Equivalence of Brody hyperbolicity and normality of quasiregular curves.
We show that for a closed n-manifold N admitting a quasiregular mapping from the Euclidean n-space the following are equivalent: (1) order of growth of π1(N) is n, (2) N is aspherical, and (3) π1(N) is virtually Zn and torsion free.
The study connects curves and cohomology on manifolds.
problem Understanding quasiregular curves and their cohomology.
method Analyzing de Rham cohomology and Künneth ideals.
result Non-trivial algebra homomorphisms between manifold cohomology and exterior algebra.
Study the exponential map on surfaces using fluid dynamics.
problem Exponential map of volume-preserving diffeomorphisms on closed surfaces.
method Fluid dynamical proof of Ebin--Misiołek--Preston theorem and extension of Shnirelman's rigidity result.
result Exponential map is a nonlinear Fredholm mapping of index zero and Fredholm quasiregular.
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or 1-quasiregular mapping between two manifolds with Cr metric tensors (r>1) is a Cr+1 conformal (local) diffeomorphism. …
Let Z be an Alexandrov space with curvature bounded below by -1 such that Z is homotopy equivalent to a real hyperbolic manifold M. It is known that the volume of Z is not smaller than the volume of M. If the volumes are equal, this short paper proves that the homotopy equivalence is homotopic to an isometric homeomorp…
The paper extends Liouville's theorem to calibrated geometries in various dimensions.
problem Extending Liouville's theorem to calibrated geometries in different dimensions.
method Analyzing Sobolev mappings and calibrations in calibrated geometries.
result Calibrations in certain dimensions have the Liouville property.
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
The affine-additive group is hyperbolic with a non-vanishing 4-capacity.
problem Characterizing the hyperbolicity of the affine-additive group.
method Proving local 4-Ahlfors regularity and hyperbolicity using a left-invariant metric and measure.
result The affine-additive group is hyperbolic with a non-vanishing 4-capacity.
In a recent preprint, Chi Li proved that aymptotically conical complex manifolds with regular tangent cone at infinity admit holomorphic compactifications (his result easily extends to the quasiregular case). In this short note, we show that if the open manifold is Calabi-Yau, then Chi Li's compactification is projecti…
This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth su…
New topological obstructions found for elliptic and quasiregularly elliptic manifolds.
problem Connection between ellipticity and quasiregular ellipticity.
method Proved new topological obstructions.
result Closed manifolds of both types must have virtually abelian fundamental group.
The article explores the mapping class group using unicellular maps and provides filtrations.
problem Understanding the structure of the mapping class group.
method Using unicellular maps and surgeries, the article describes the mapping class group.
result Provides filtrations of the mapping class group.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
Deep learning classifies seven types of maps for better access.
problem Efficiently accessing the right map type from digital maps.
method Used deep convolutional neural networks to classify seven types of maps.
result Deep learning can accurately classify different types of maps.
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.
Both bi-harmonic map and f-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f-bi-harmonic maps as the critical points of the f-bi-energy functional 21∫Mf∣τ(φ)∣2dvg. This class of maps generalizes both …
Generic pseudo-Anosov mapping classes in mapping class groups.
problem Understanding the prevalence of pseudo-Anosov mapping classes.
method Proving genericity with respect to specific notions of genericity.
result Pseudo-Anosov mapping classes are generic in mapping class groups.
Paper constructs maps for sutured monopole Floer homology.
problem None explicitly stated in the abstract.
method Constructs gluing and cobordism maps for sutured monopole Floer homology.
result Developed mathematical tools for sutured monopole Floer homology.