EXTREMAL CONFORMAL MAPPINGS AND POLES OF QUADRATIC DIFFERENTIALS

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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RECURSION FOR MASUR-VEECH VOLUMES OF MODULI SPACES OF QUADRATIC DIFFERENTIALS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748020000638 ◽

2021 ◽

pp. 1-6

Author(s):

Maxim Kazarian

Keyword(s):

Moduli Spaces ◽

Recursion Relation ◽

Quadratic Differentials ◽

Hodge Integrals

Abstract We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

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A Purely Geometrical Introduction of Spinors in Special Relativity by Means of Conformal Mappings on the Celestial Sphere

Annalen der Physik ◽

10.1002/andp.19734850303 ◽

1973 ◽

Vol 485 (3-4) ◽

pp. 206-210 ◽

Cited By ~ 3

Author(s):

D. W. Ebner

Keyword(s):

Special Relativity ◽

Conformal Mappings ◽

Celestial Sphere

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Conformal mapping of the Misner–Sharp mass from gravitational collapse

International Journal of Modern Physics D ◽

10.1142/s0218271816500814 ◽

2016 ◽

Vol 25 (07) ◽

pp. 1650081 ◽

Cited By ~ 9

Author(s):

Fayçal Hammad

Keyword(s):

Conformal Mapping ◽

Gravitational Collapse ◽

Conformal Transformation ◽

Conformal Mappings ◽

Conformal Transformations ◽

Theories Of Gravity ◽

Definition Of ◽

Geometric Definition

The conformal transformation of the Misner–Sharp mass is reexamined. It has recently been found that this mass does not transform like usual masses do under conformal mappings of spacetime. We show that when it comes to conformal transformations, the widely used geometric definition of the Misner–Sharp mass is fundamentally different from the original conception of the latter. Indeed, when working within the full hydrodynamic setup that gave rise to that mass, i.e. the physics of gravitational collapse, the familiar conformal transformation of a usual mass is recovered. The case of scalar–tensor theories of gravity is also examined.

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