THE WEIERSTRASS E-FUNCTION IN DIFFERENTIAL METRIC GEOMETRY

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

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Metric geometry of partial isometries in a finite von Neumann algebra

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2007.04.056 ◽

2008 ◽

Vol 337 (2) ◽

pp. 1226-1237 ◽

Cited By ~ 1

Author(s):

Esteban Andruchow

Keyword(s):

Von Neumann Algebra ◽

Metric Geometry ◽

Partial Isometries ◽

Von Neumann ◽

Finite Von Neumann Algebra

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On the Metric Geometry of the Plane N-Line

Transactions of the American Mathematical Society ◽

10.2307/1986387 ◽

1900 ◽

Vol 1 (2) ◽

pp. 97 ◽

Cited By ~ 1

Author(s):

F. Morley

Keyword(s):

Metric Geometry

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Fourier Integrals and Metric Geometry

I.J. Schoenberg Selected Papers ◽

10.1007/978-1-4612-3946-8_10 ◽

1988 ◽

pp. 146-171

Author(s):

J. Von Neumann ◽

I. J. Schoenberg

Keyword(s):

Metric Geometry ◽

Fourier Integrals

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The metric geometry of near equilibrium irreversible thermodynamics

The Journal of Chemical Physics ◽

10.1063/1.439545 ◽

1980 ◽

Vol 72 (5) ◽

pp. 3127-3129

Author(s):

Gilbert Nathanson ◽

Oktay Sinanoğlu

Keyword(s):

Irreversible Thermodynamics ◽

Metric Geometry

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COARSE COHERENCE OF METRIC SPACES AND GROUPS AND ITS PERMANENCE PROPERTIES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000977 ◽

2018 ◽

Vol 98 (3) ◽

pp. 422-433

Author(s):

BORIS GOLDFARB ◽

JONATHAN L. GROSSMAN

Keyword(s):

Metric Spaces ◽

General Context ◽

Metric Geometry ◽

Finitely Generated ◽

Coherence Property ◽

Finitely Generated Groups ◽

Permanence Property ◽

Permanence Properties ◽

New Framework

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

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