"Teichmuller theory"의 두 판 사이의 차이

2021년 2월 17일 (수) 02:32 기준 최신판

introduction

review of hyperbolic geometry

horocycle
- http://en.wikipedia.org/wiki/Horocycle
- http://web1.kcn.jp/hp28ah77/us15_horo.htm
exponentiated hyperbolic distances between horocycles drawn around vertices of a polygon with geodesic sides and cusps at the vertices
lamination
shear coordinates
lambda length
http://moniker.name/worldmaking/?p=744
http://orion.math.iastate.edu/dept/thesisarchive/MSCC/OLearyMSCCSS06.pdf
\def
An ideal triangle in (S,M) is a triangle with vertices at M, whose sides are geodesics.
\def
A horocycle at marked point p is a set of points "equidistant" to p. In lift to H^2, looks like circle tangent to boundary at p.

Teichmuller space of a marked surface

Given marked surface (S,M) , the Teichmuller space T(S,M) is the space of metrics on (S,M) such that

are hyperbolic (constant curvature -1)
have geodesic boundary at boundary of S
local neighborhood of point on boundary S can be mapped isometrically to neighborhood of a point here on one side of geodesic
have cusps at points in M

Considered up to diffeomorphism homotopic to identity.

Facts

(1) T(S,M) contractible

(2) T(S,M) is manifold of dimension 6g-6+2p+3b+c where g = genus, p=# of puncture, b = # boundary component, c=# of marked points on boundary

related items

Moduli space of local systems and higher Teichmuller theory

expositions

Norbert A'Campo, Lizhen Ji, Athanase Papadopoulos, On Grothendieck's construction of Teichmüller space, http://arxiv.org/abs/1603.02229v1
Matheus, Carlos. “Lecture Notes on the Dynamics of the Weil-Petersson Flow.” arXiv:1601.00690 [math], January 4, 2016. http://arxiv.org/abs/1601.00690.
Papadopoulos, Athanase, Vincent Alberge, and Weixu Su. “A Commentary on Teichm"uller’s Paper ‘Extremale Quasikonforme Abbildungen Und Quadratische Differentiale.’” arXiv:1511.01313 [math], November 4, 2015. http://arxiv.org/abs/1511.01313.
Introduction to Teichmüller theory, old and new, Athanase Papadopoulos

articles

Leonid Chekhov, Marta Mazzocco, Colliding holes in Riemann surfaces and quantum cluster algebras, arXiv:1509.07044 [math-ph], September 23 2015, http://arxiv.org/abs/1509.07044
Lien-Yung Kao, Pressure type metrics on spaces of metric graphs, arXiv:1604.03173 [math.DS], April 11 2016, http://arxiv.org/abs/1604.03173
Babak Modami, Asymptotics of a class of Weil-Petersson geodesics and divergence of Weil-Petersson geodesics, Algebr. Geom. Topol. 16 (2016) no.1, pp. 267-323, http://arxiv.org/abs/1401.3234v4
Antonakoudis, Stergios M. “The Complex Geometry of Teichm"uller Spaces and Bounded Symmetric Domains.” arXiv:1510.07340 [math], October 25, 2015. http://arxiv.org/abs/1510.07340.
Penner, R. C., and Anton M. Zeitlin. “Decorated Super-Teichm"uller Space.” arXiv:1509.06302 [hep-Th, Physics:math-Ph], September 21, 2015. http://arxiv.org/abs/1509.06302.

메타데이터

위키데이터

ID : Q2400539

Spacy 패턴 목록

[{'LOWER': 'teichmüller'}, {'LEMMA': 'space'}]

@@ 1번째 줄: / 1번째 줄: @@
-<h5>introduction</h5>
+==introduction==
-<h5>Teichmuller space o</h5>
+==review of hyperbolic geometry==
+*  horocycle
+** http://en.wikipedia.org/wiki/Horocycle
+** http://web1.kcn.jp/hp28ah77/us15_horo.htm
+* exponentiated hyperbolic distances between horocycles drawn around vertices of a polygon with geodesic sides and cusps at the vertices
+* lamination
+* shear coordinates
+* lambda length
+* http://moniker.name/worldmaking/?p=744
+* http://orion.math.iastate.edu/dept/thesisarchive/MSCC/OLearyMSCCSS06.pdf
+* \def
+* An ideal triangle in (S,M) is a triangle with vertices at M, whose sides are geodesics.
+* \def
+* A horocycle at marked point p is a set of points "equidistant" to p. In lift to H^2, looks like circle tangent to boundary at p.
+==Teichmuller space of a marked surface==
 Given marked surface (S,M) , the Teichmuller space T(S,M) is the space of metrics on (S,M) such that
 * are hyperbolic  (constant curvature -1)
 * have geodesic boundary at boundary of S
 * local neighborhood of point on boundary S can be mapped isometrically to neighborhood of a point here on one side of geodesic
-* have cusps at points in M
+* have cusps at points in M
 Considered up to diffeomorphism homotopic to identity.
@@ 24번째 줄: / 42번째 줄: @@
 (2) T(S,M) is manifold of dimension 6g-6+2p+3b+c where g = genus, p=# of puncture, b = # boundary component, c=# of marked points on boundary
-\def
-An ideal triangle in (S,M) is a triangle with vertices at M, whose sides are geodesics.
+==related items==
+* [[Moduli space of local systems and higher Teichmuller theory]]
-\def
-A horocycle at marked point p is a set of points "equidistant" to p. In lift to H^2, looks like circle tangent to boundary at p.
+==expositions==
+* Norbert A'Campo, Lizhen Ji, Athanase Papadopoulos, On Grothendieck's construction of Teichmüller space, http://arxiv.org/abs/1603.02229v1
+* Matheus, Carlos. “Lecture Notes on the Dynamics of the Weil-Petersson Flow.” arXiv:1601.00690 [math], January 4, 2016. http://arxiv.org/abs/1601.00690.
+* Papadopoulos, Athanase, Vincent Alberge, and Weixu Su. “A Commentary on Teichm"uller’s Paper ‘Extremale Quasikonforme Abbildungen Und Quadratische Differentiale.’” arXiv:1511.01313 [math], November 4, 2015. http://arxiv.org/abs/1511.01313.
-\def decorated Teichmuller space \tilde{T}(S,M) is
+* [http://www.ems-ph.org/books/055/9783037190296_introduction.pdf Introduction to Teichmüller theory, old and new], Athanase Papadopoulos
-* a point in T(S,M)
+==articles==
-* a choic of horocycle around each cusp from M
+* Leonid Chekhov, Marta Mazzocco, Colliding holes in Riemann surfaces and quantum cluster algebras, arXiv:1509.07044 [math-ph], September 23 2015, http://arxiv.org/abs/1509.07044
+* Lien-Yung Kao, Pressure type metrics on spaces of metric graphs, arXiv:1604.03173 [math.DS], April 11 2016, http://arxiv.org/abs/1604.03173
+* Babak Modami, Asymptotics of a class of Weil-Petersson geodesics and divergence of Weil-Petersson geodesics, Algebr. Geom. Topol. 16 (2016) no.1, pp. 267-323, http://arxiv.org/abs/1401.3234v4
+* Antonakoudis, Stergios M. “The Complex Geometry of Teichm"uller Spaces and Bounded Symmetric Domains.” arXiv:1510.07340 [math], October 25, 2015. http://arxiv.org/abs/1510.07340.
+* Penner, R. C., and Anton M. Zeitlin. “Decorated Super-Teichm"uller Space.” arXiv:1509.06302 [hep-Th, Physics:math-Ph], September 21, 2015. http://arxiv.org/abs/1509.06302.
-\def (Penner) For a arc A in (S,M) and \Sigma\in\tilde{T}(S,M),
-the length of A with respect to \Sigma is
-l_{\Sigma(A) = length on geodesic representative of A between intersections with horocycles around ends. (negative if 2 horocycles intersect)
+[[분류:개인노트]]
+[[분류:cluster algebra]]
+[[분류:math and physics]]
+[[분류:math]]
+[[분류:migrate]]
+==메타데이터==
+===위키데이터===
-The \lambda - length is \lambda_{\Sigma}(A) : = e^{l_{\Sigma}(A)/2}
+* ID :  [https://www.wikidata.org/wiki/Q2400539 Q2400539]
+===Spacy 패턴 목록===
-Note  :  \lambda_{\Sigma}(A) in \mathbb{R}_{> 0 }
+* [{'LOWER': 'teichmüller'}, {'LEMMA': 'space'}]
-\prop
-In an ideal quadrilateral, the Ptolemy relation holds.
-\thm (Penner)
-For any triangulation (A_{i})
-* [http://www.ems-ph.org/books/055/9783037190296_introduction.pdf Introduction to Teichmüller theory, old and new], Athanase Papadopoulos