New mappings on closed manifolds can't be broken down easily.
problem Existence of indecomposable quasiconformal maps.
method Demonstrated existence through mappings on closed manifolds.
result Indecomposable quasiconformal maps exist on closed manifolds.
Optimal interpolation schemes minimize quasiconformal distortion of projective transformations.
problem Minimizing distortion in projective transformations and discrete conformal maps.
method Analysis of quasiconformal dilatation and comparison of interpolation schemes.
result Angle bisector preserving interpolation is optimal in terms of dilatation.
New metric defines surface shapes, minimizing area and angle distortions.
problem Defining and measuring the shape of high genus surfaces.
method Defined a metric space, introduced energies for area and angle distortions, showed minimizers by lower semicontinuity.
result Energy minimizers in surface shape space correspond to quasiconformal homeomorphisms.
Fast algorithm for spherical parameterization of surfaces with adaptive remeshing.
problem Efficiently parameterizing genus-0 closed surfaces with user-defined quasiconformal distortion.
method Proposes a fast algorithm for spherical quasiconformal parameterization.
result Effective for adaptive surface remeshing in computer graphics and animations.
The paper constructs quasiconformal mappings in the Heisenberg group.
problem Constructing quasiconformal mappings in the Heisenberg group that minimize a mean distortion functional.
method Constructing a corresponding problem in the Poincaré half-plane and using geometric conditions to find the mappings.
result The method provides a unique way to construct minimizers of the mean distortion functional.
New stretch maps minimize distortion in geometric group theory.
problem Finding optimal maps in geometric group theory.
method Proving minimizers using modulus of curve families and MSP.
result Stretch maps are minimizers of mean quasiconformal distortion.
Method flattens complex surfaces with consistent density and shape.
problem Shape deformations and local geometric distortions in density-equalizing maps for multiply-connected surfaces.
method Formulates density diffusion as a quasiconformal flow, solving an energy minimization problem involving the Beltrami coefficient to ensure bijectivity and control distortion.
result Achieves optimal parameterization of multiply-connected surfaces with bijective and controlled geometric distortions.
Optimizes maps with controlled distortion for geometric tasks.
problem Free-boundary diffeomorphism optimization in geometric modeling.
method Least-squares quasiconformal (LSQC) operator and Spectral Beltrami Network (SBN).
result LSQC minimizer well-posed under mild conditions, stable under mesh refinement.
Tissot's indicatrix theory is foundational for quasiconformal mappings.
problem Understanding map distortions in geographical projections.
method Mathematical analysis of map projections and their distortions.
result Tissot's work laid the groundwork for quasiconformal mappings.
It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity …
The study examines Teichmüller distances in lattices and punctured tori.
problem Distribution of Teichmüller distances in moduli spaces.
method Uniform distribution of lattices, calculation of Teichmüller distances.
result Identifies distribution of distances in Teichmüller space.
By the Riemann-mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of P to those Q. In this case, one wants to find the ``best" mapping between these polygons, i.e.…
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.
In a very influential paper Gehring and Palka introduced the notions of quasiconformally homogeneous and uniformly quasiconformally homogeneous subsets of Euclidean space. Their motivation was to provide a characterization of quasi-disks, i.e. domains which are quasiconformally homeomorphic to the unit disk. As a gener…
Study shows Julia sets and gasket limit sets are quasiconformally different.
problem Quasiconformal non-equivalence of Julia sets and gasket limit sets.
method Proved quasiconformal non-equivalence of Julia sets and gasket limit sets.
result Julia sets and gasket limit sets are quasiconformally different.
A closed hyperbolic Riemann surface M is said to be K-quasiconformally homogeneous if there exists a transitive family F of K-quasiconformal homeomorphisms. Further, if all [f] in F act trivially on H1(M;Z), we say M is Torelli-K-quasiconformally homogeneous. We prove the existence of a uniform lower bound on K for Tor…
Study lifts plane mappings to Heisenberg group.
problem Contact quasiconformal mappings in hyperbolic Heisenberg group.
method Lifting Theorem for symplectic mappings.
result Symplectic mappings lifted to Heisenberg group.
Explains Teichmüller's 1944 displacement theorem for quasiconformal mappings.
problem Finding the quasiconformal mapping with minimal dilatation.
method Detailed explanation of Teichmüller's solution.
result Solution to extremal problem in quasiconformal mappings.
We consider quasiconformal deformations of C∖Z. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to C∖Z. In particular, we characterize the closed subsets of R whose compliments are quasiconformally equivalen…
Characterizes quasiconformally homogeneous ladder surfaces.
problem Characterize quasiconformally homogeneous Riemann surfaces.
method Breaks the problem into four cases and proves one case.
result Every quasiconformally homogeneous ladder surface is quasiconformally equivalent to a regular cover of a closed surface.
This study extends a result on quasi-isometry of hyperbolic groups to relatively hyperbolic groups.
problem Classifying groups up to quasi-isometry, focusing on relatively hyperbolic groups.
method Defining quasiconformal maps and showing their equivalence to coarsely cusp-preserving quasi-isometries between Bowditch boundaries.
result Quasiconformal maps between Bowditch boundaries of relatively hyperbolic groups are equivalent to coarsely cusp-preserving quasi-isometries.
We provide new conditions that ensure that two metric measure spaces are not quasiconformally equivalent. As an application we deduce that there exists no quasiconformal map between the sub-Riemannian Heisenberg and roto-translation groups.
Characterizes quasiconformal homeomorphisms on surfaces.
problem Understanding the group of quasiconformal homeomorphisms on surfaces.
method Combinatorial characterization of quasiconformal homeomorphisms via graphs of essential quasicircles.
result Quasiconformal homeomorphisms are automorphisms of a graph of essential quasicircles on a surface.
For the result on 1-quasiconformal maps, see the paper by Cowling and Ottazzi. The result on quasiconformal maps on Carnot groups with reducible first layer will appear in a forthcoming paper by Enrico Le Donne and Xiangdong Xie.
Overview of quasiconformal mappings on Heisenberg group.
problem Understanding quasiconformal mappings on the Heisenberg group.
method Examination of Korányi-Reimann theory and extensions.
result Comparison of Heisenberg group with classical two-dimensional case.
A Riemann surface M is said to be K-quasiconformally homogeneous if for every two points p,q∈M, there exists a K-quasiconformal homeomorphism f:M→M such that f(p)=q. In this paper, we show there exists a universal constant K0>1 such that if M is a K-quasiconformally homogen…
In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic…
Teichmuller proved the existence of extremal quasiconformal mappings for pentagons.
problem Existence of extremal quasiconformal mappings for pentagons.
method Proof of existence for quasiconformal mappings in the case of pentagons.
result Existence of extremal quasiconformal mappings for pentagons.
Generalizes Frobenius theorem to quasiconformal deformations.
problem Integrability of plane fields generated by quasiconformal deformations.
method Generalization of classical Frobenius theorem to CQ plane fields. result A.e. involutive CQ plane fields are integrable. Derives conditions for embedding smooth surfaces into higher dimensions.
problem Conditions for embedding smooth surfaces into higher dimensions.
method Derives necessary and sufficient conditions for 1-quasiconformal parameterization.
result Liouville theorem does not extend to embeddings of domains into higher dimensions.
Research connects geometry and sound frequencies of shapes.
problem Linking geometric shapes to their sound frequencies.
method Analyzes algebra of multipliers and spectral properties of Dirichlet integral.
result Proves homeomorphisms preserving fundamental tones in Euclidean domains.
Harmonic extension of Weil-Petersson circle homeomorphisms
problem Harmonic maps from the Weil-Petersson disk to the hyperbolic disk
method Anti-holomorphic L2-energy result Harmonic extension of Weil-Petersson circle homeomorphisms minimizes the L2-energy Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.
problem Finding compact surfaces in Euclidean 3-space with specific curvature properties.
method Representation of solutions to linear elliptic equations with discontinuous coefficients.
result Compact surfaces of genus zero with specific curvature properties are round spheres.
We introduce the notion of a conformal de Rham complex of a Riemannian manifold. This is a graded differential Banach algebra and it is invariant under quasiconformal maps, in particular the associated cohomology is a new quasiconformal invariant.
We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…
In this paper we construct quasiconformal embeddings from Y-pieces that contain a short boundary geodesic into degenerate ones. These results are used in a companion paper to study the Jacobian tori of Riemann surfaces that contain small simple closed geodesics.
We characterize the rigidity of Carnot groups in the class of C2 contact maps in terms of complex characteristics. Furthermore, we obtain a Liouville type theorem for Carnot groups which states that 1-quasiconformal maps form finite dimensional Lie groups.
Grötzsch's papers review progress in quasiconformal geometry.
problem Developing quasiconformal mappings theory.
method Analyzing five papers by Grötzsch from 1928-1932.
result Illustrates Grötzsch's motivation and results on quasiconformal mappings.
Countable modular groups found on surfaces with infinite type.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
Commentary on Teichmüller's 1938 paper on conformal and quasiconformal mappings.
problem Investigations into conformal and quasiconformal mappings and their applications.
method Detailed development of conformal invariants and applications in value distribution theory.
result Insures the almost circularity of certain loci and the circularity near infinity of quasiconformal maps.
Study on curvature conditions for non-conformally flat spheres using quasiconformal maps and Ricci flow.
problem Curvature conditions for non-conformally flat spheres.
method Construct quasiconformal maps and apply Ricci flow.
result Controlled bilipschitz constant between metrics.
A closed discrete subset A⊂C is called tame if C∖A is quasiconformally equivalent to C∖Z. By giving several criteria for A to be tame, we shall show that Z+iZ is not tame.
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.
Survey of quasiconformal mappings from ancient cartography to Teichmüller.
problem Finding mappings with minimal deviation from conformality.
method Historical survey and analysis of key mathematicians' works.
result Introduction of Tissot's work on infinitesimal ellipses.
Locally connected deformation spaces for 3-manifolds.
problem Locating quasiconformally rigid points in hyperbolic 3-manifolds.
method Proving local connectedness at specific points in the deformation space.
result The deformation space is locally connected at quasiconformally rigid points.
Existence and rigidity results for lifts in Carnot groups.
problem Existence and properties of lifts for maps between Carnot groups.
method Use central extensions to define lifts and prove existence and rigidity results for Lipschitz, Sobolev, and quasiconformal maps.
result Quasiconformal maps admit contact lifts that are bi-Lipschitz.
New methods study heat flow convergence on hyperbolic spaces.
problem Convergence of heat flow on negatively curved spaces.
method Developed new methods for studying convergence of heat flow.
result Proved extension of quasiconformal maps to hyperbolic spaces.
This paper is devoted to the study of the global properties of harmonically immersed Riemann surfaces in R3. We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those with finite total curvature. We pay special attention to the construction of new ex…