Local boundary dilatation of quasiconformal maps

2005 ◽  
Vol 48 (6) ◽  
pp. 798
Author(s):  
Yi QI
Keyword(s):  
Quasiconformal Maps ◽  
Local Boundary

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

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.

scholarly journals Replica wormholes for an evaporating 2D black hole

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Kanato Goto ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Path Integral ◽  
Island Rule ◽  
Technical Challenge ◽  
Local Boundary ◽  
Lorentzian Signature ◽  
Special Cases ◽  
Non Local ◽  
Non Local Equations

Abstract Quantum extremal islands reproduce the unitary Page curve of an evaporating black hole. This has been derived by including replica wormholes in the gravitational path integral, but for the transient, evaporating black holes most relevant to Hawking’s paradox, these wormholes have not been analyzed in any detail. In this paper we study replica wormholes for black holes formed by gravitational collapse in Jackiw-Teitelboim gravity, and confirm that they lead to the island rule for the entropy. The main technical challenge is that replica wormholes rely on a Euclidean path integral, while the quantum extremal islands of an evaporating black hole exist only in Lorentzian signature. Furthermore, the Euclidean equations for the Schwarzian mode are non-local, so it is unclear how to connect to the local, Lorentzian dynamics of an evaporating black hole. We address these issues with Schwinger-Keldysh techniques and show how the non-local equations reduce to the local ‘boundary particle’ description in special cases.

Structural development of the Snow Lake Allochthon and its role in the evolution of the southeastern Trans-Hudson Orogen in Manitoba, central Canada

1999 ◽  
Vol 36 (11) ◽  
pp. 1881-1899 ◽  
Author(s):  
Jürgen Kraus ◽  
Paul F Williams
Keyword(s):  
Structural Development ◽  
Southeastern Part ◽  
External Zone ◽  
Local Boundary ◽  
Tectonic Model ◽  
Adjacent Part ◽  
Central Canada ◽  
Tectonic Transport ◽  
The One ◽  
Superior Craton

The Snow Lake Allochthon is a zone of tectonic interleaving of sedimentary rocks of an inverted marginal basin (Kisseynew Domain) with island-arc and oceanic rocks. It is located in the southeastern part of the exposed internal zone of the Paleoproterozoic Trans-Hudson Orogen in Manitoba, Canada, near the external zone (Superior collision zone or Thompson Belt), which constitutes the local boundary between the Trans-Hudson Orogen and the Archean Superior Craton. The Snow Lake Allochthon formed, was deformed, and was metamorphosed up to high grade at low to medium pressure during the Hudsonian orogeny as a result of the collision of Archean cratons ~1.84-1.77 Ga. Four generations of folds (F1-F4) that formed in at least three successive kinematic frames over a period of more than 30 Ma are described. Isoclinal to transposed southerly verging F1-2 structures are refolded by large, open to tight F3 folds and, locally, by open to tight F4 folds. The axes of the F1-2 folds are parallel or near parallel to the axes of F3 folds, owing to progressive reorientation of the F1-2 axes during south- to southwest-directed tectonic transport, followed by F3 refolding around the previous linear anisotropy. A tectonic model is presented that reconciles the distinct tectono-metamorphic developments in the Snow Lake Allochthon and the adjacent part of the Kisseynew Domain on the one hand, and in the Thompson Belt on the other, during final collision of the Trans-Hudson Orogen with the Superior Craton.

Sign in / Sign up

Export Citation Format

Share Document