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Geometric branched covers between generalized manifolds
Juha Heinonen,Seppo Rickman +1 more
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In this paper, a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds is developed, and a construction that generalizes an extension theorem for BRC maps by I. Berstein and A. Edmonds is given.Abstract:
We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension.read more
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Lectures on Lipschitz analysis
TL;DR: The theory of Lipschitz functions in Euclidean spaces was studied in this paper, where it was shown that a function is a Lipschnitz function if it is L-Lipschnitzer for some L.
Journal ArticleDOI
Uniformization of two-dimensional metric surfaces
TL;DR: In this article, the authors give necessary and sufficient conditions for Ahlfors 2-regular spheres by quasisymmetric maps to be equivalent to the Euclidean plane, disk, or sphere.
Journal ArticleDOI
Hyperelastic Deformations of Smallest Total Energy
Tadeusz Iwaniec,Jani Onninen +1 more
TL;DR: In this paper, the authors considered the problem of finding suitable free Lagrangians and using them for a specific stored energy function E in the plane, assuming that E is conformally coerced and polyconvex, and established the existence and global invertibility of the minimizers.
Journal ArticleDOI
On the locally branched Euclidean metric gauge
Juha Heinonen,Dennis Sullivan +1 more
TL;DR: In this paper, a mixture of analysis, geometry, and topology with an argument of Yu. Reshetnyak to produce the branched coordinates for the gauge is presented.
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Sharpness of Rickman's Picard theorem in all dimensions
David Drasin,Pekka Pankka +1 more
TL;DR: In this article, it was shown that given a finite set of points and a quasiregular mapping, there exists a quasi-global mapping of points from a given set to a set of missing points.
References
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Geometric Measure Theory
TL;DR: In this article, Grassmann algebras of a vectorspace have been studied in the context of the calculus of variations, and a glossary of some standard notations has been provided.
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Lectures on Algebraic Topology
TL;DR: In this paper, the authors provide a preliminary introduction to the categories, Abelian groups, and homotopy of complexes in the Euclidean space, and a discussion of the application of these categories to Euclideans.
Journal ArticleDOI
Differentiability of Lipschitz Functions on Metric Measure Spaces
TL;DR: In this paper, the authors propose a method to solve the problem of the problem: without abstracts, without abstractions, without Abstracts. (Without Abstract) (without Abstract)
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Sobolev met Poincaré
Piotr Hajłasz,Pekka Koskela +1 more
TL;DR: In this article, the Poincare and Sobolev inequalities, pointwise estimates, and pointwise classifications of Soboleve classes are discussed. But they do not cover the necessary conditions for Poincarse inequalities.
Book
Lectures on n-Dimensional Quasiconformal Mappings
TL;DR: The modulus of a curve family and the analytic properties of quasiconformal mappings have been studied in real analysis as discussed by the authors, where the modulus is defined as a function of the curve family modulus.