Teichmuller theory

 

 

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

 

 

 

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

 

 

\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.

 

 

 

decorated Teichmuller space

\def decorated Teichmuller space \tilde{T}(S,M) is

• a point in T(S,M)
• a choic of horocycle around each cusp from M

\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) 

 

The \lambda - length is \lambda_{\Sigma}(A) : = e^{l_{\Sigma}(A)/2}

Note  :  \lambda_{\Sigma}(A) in \mathbb{R}_{> 0 }

 

\prop

In an ideal quadrilateral, the Ptolemy relation holds.

 

\thm (Penner)

For any triangulation (A_{i})

 

 

shear coordinate

 

• for tropical version, see lamination and shear coordinates on a marked surface

 

 

history

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

 

 

 

 

 

 

 

encyclopedia

 

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://eom.springer.de
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

 

 

 

books

 

 

 

• 2011년 books and articles
• http://library.nu/search?q=
• http://library.nu/search?q=

 

 

 

 

 

expositions

 

• Introduction to Teichmüller theory, old and new, Athanase Papadopoulos

 

 

 

 

articles

 

•  
  
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/

 

 

 

 

 

question and answers(Math Overflow)

 

• http://mathoverflow.net/search?q=
• http://math.stackexchange.com/search?q=
• http://physics.stackexchange.com/search?q=

 

 

 

 

 

 

 

blogs

 

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

 

 

 

experts on the field

 

• http://arxiv.org/

 

 

 

 

 

links

 

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