[PDF] Preservation Of Bounded Geometry Under Transformations Of Metric Spaces eBook

Author : Xining Li
Publisher :
Page : 138 pages
File Size : 44,72 MB
Release : 2015
Category :
ISBN :

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In the theory of geometric analysis on metric measure spaces, two properties of a metric measure space make the theory richer. These two properties are the doubling property of the measure, and the support of a Poincare ́inequality by the metric measure space. The focus of this dissertation is to show that the doubling property of the measure and the support of a Poincare ́ inequality are preserved by two transformations of the metric measure space: sphericalization (to obtain a bounded space from an unbounded space), and flattening (to obtain an unbounded space from a bounded space). We will show that if the given metric measure space is equipped with an Ahlfors Q-regular measure, then so are the spaces obtained by the sphericalization/flattening transformations. We then show that even if the measure is not Ahlfors regular, if it is doubling, then the transformed measure is still doubling. We then show that if the given metric space satisfies an annular quaisconvexity property and the measure is doubling, and in addition if the metric measure space supports a p-Poincare ́inequality in the sense of Heinonen and Koskela's theory, then so does the transformed metric measure space (under the sphericalization/flattening procedure). Finally, we show that if we relax the annular quasiconvexity condition to an analog of the starlike condition for the metric measure space, then if the metric measure space also satisfies a p-Poincare ́inequality, the transformed space also must satisfy a q-Poincare ́inequality for some p

Author : Dmitri Burago
Publisher : American Mathematical Society
Page : 415 pages
File Size : 16,56 MB
Release : 2022-01-27
Category : Mathematics
ISBN : 1470468530

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“Metric geometry” is an approach to geometry based on the notion of length on a topological space. This approach experienced a very fast development in the last few decades and penetrated into many other mathematical disciplines, such as group theory, dynamical systems, and partial differential equations. The objective of this graduate textbook is twofold: to give a detailed exposition of basic notions and techniques used in the theory of length spaces, and, more generally, to offer an elementary introduction into a broad variety of geometrical topics related to the notion of distance, including Riemannian and Carnot-Carathéodory metrics, the hyperbolic plane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (non-positively and non-negatively curved spaces). The authors tend to work with “easy-to-touch” mathematical objects using “easy-to-visualize” methods. The authors set a challenging goal of making the core parts of the book accessible to first-year graduate students. Most new concepts and methods are introduced and illustrated using simplest cases and avoiding technicalities. The book contains many exercises, which form a vital part of the exposition.

Author : Stephanie Alexander
Publisher : Springer
Page : 88 pages
File Size : 43,14 MB
Release : 2019-05-08
Category : Mathematics
ISBN : 3030053121

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Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.

Author : Tushar Das
Publisher : American Mathematical Soc.
Page : 321 pages
File Size : 36,16 MB
Release : 2017-04-14
Category : Mathematics
ISBN : 1470434652

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This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.

Author : Christian Rosendal
Publisher : Cambridge University Press
Page : 309 pages
File Size : 18,28 MB
Release : 2021-12-16
Category : Mathematics
ISBN : 110884247X

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Provides a general framework for doing geometric group theory for non-locally-compact topological groups arising in mathematical practice.

Author : Matti Vuorinen
Publisher : Springer
Page : 228 pages
File Size : 49,70 MB
Release : 2006-11-15
Category : Mathematics
ISBN : 3540392076

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This book is an introduction to the theory of spatial quasiregular mappings intended for the uninitiated reader. At the same time the book also addresses specialists in classical analysis and, in particular, geometric function theory. The text leads the reader to the frontier of current research and covers some most recent developments in the subject, previously scatterd through the literature. A major role in this monograph is played by certain conformal invariants which are solutions of extremal problems related to extremal lengths of curve families. These invariants are then applied to prove sharp distortion theorems for quasiregular mappings. One of these extremal problems of conformal geometry generalizes a classical two-dimensional problem of O. Teichmüller. The novel feature of the exposition is the way in which conformal invariants are applied and the sharp results obtained should be of considerable interest even in the two-dimensional particular case. This book combines the features of a textbook and of a research monograph: it is the first introduction to the subject available in English, contains nearly a hundred exercises, a survey of the subject as well as an extensive bibliography and, finally, a list of open problems.

Author : Owen Dearricott
Publisher : Springer
Page : 202 pages
File Size : 41,14 MB
Release : 2014-07-22
Category : Mathematics
ISBN : 3319063731

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Providing an up-to-date overview of the geometry of manifolds with non-negative sectional curvature, this volume gives a detailed account of the most recent research in the area. The lectures cover a wide range of topics such as general isometric group actions, circle actions on positively curved four manifolds, cohomogeneity one actions on Alexandrov spaces, isometric torus actions on Riemannian manifolds of maximal symmetry rank, n-Sasakian manifolds, isoparametric hypersurfaces in spheres, contact CR and CR submanifolds, Riemannian submersions and the Hopf conjecture with symmetry. Also included is an introduction to the theory of exterior differential systems.

Author : Boris Khesin
Publisher : Springer Science & Business Media
Page : 304 pages
File Size : 40,19 MB
Release : 2008-09-28
Category : Mathematics
ISBN : 3540772634

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This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.

Author : Jared Ross Bunn
Publisher :
Page : 55 pages
File Size : 36,53 MB
Release : 2011
Category :
ISBN :

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We begin by recalling the notion of a coarse space as dened by John Roe. We show that metrizability of coarse spaces is a coarse invariant. The concepts of bounded geometry, asymptotic dimension, and Guoliang Yu's Property A are investigated in the setting of coarse spaces. In particular, we show that bounded geometry is a coarse invariant, and we give a proof that nite asymptotic dimension implies Property A in this general setting. The notion of a metric approximation is introduced, and a characterization theorem is proved regarding bounded geometry. Chapter 7 presents a discussion of coarse structures on the minimal uncountable ordinal. We show that it is a nonmetrizable coarse space not of bounded geometry. Moreover, we show that this space has asymptotic dimension 0; hence, it has Property A. Finally, Chapter 8 regards coarse structures on products of coarse spaces. All of the previous concepts above are considered with regard to 3 dierent coarse structures analogous to the 3 dierent topologies on products in topology. In particular, we see that an arbitrary product of spaces with any of the 3 coarse structures with asymptotic dimension 0 has asymptotic dimension 0.

Author : Vadim Kaimanovich
Publisher : Walter de Gruyter
Page : 545 pages
File Size : 28,77 MB
Release : 2008-08-22
Category : Mathematics
ISBN : 3110198088

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Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik.