"Teichmuller theory"의 두 판 사이의 차이
| 48번째 줄: | 48번째 줄: | ||
<h5>shear coordinate</h5> | <h5>shear coordinate</h5> | ||
| − | * | + | * E an edge (diagonal), T a triangulation (collection of edges or diagonals) |
| + | * <math>\Sigma</math> a point in the Teichmuller space (metric) | ||
| + | * shear coordinate <math>\tau_{\Sigma}(E,T)</math> | ||
* for tropical version of shear coordiante, see [[lamination and tropical shear coordinates on a marked surface|lamination and shear coordinates on a marked surface]] | * for tropical version of shear coordiante, see [[lamination and tropical shear coordinates on a marked surface|lamination and shear coordinates on a marked surface]] | ||
| 59번째 줄: | 61번째 줄: | ||
<h5>decorated Teichmuller space</h5> | <h5>decorated Teichmuller space</h5> | ||
| − | \def decorated Teichmuller | + | \def decorated Teichmuller space <math>\tilde{T}(S,M)</math> is |
| − | * a point in T(S,M) | + | * a point in T(S,M) (i.e. hyperbolic metric) |
| − | * a choic of horocycle around each cusp from M | + | * a choic of horocycle around each cusp from M (i.e. marked point) |
| − | \def (Penner) For a arc A in (S,M) and \Sigma\in\tilde{T}(S,M), | + | \def (Penner) For a arc A in (S,M) and <math>\Sigma\in\tilde{T}(S,M)</math>, |
the length of A with respect to \Sigma is | the length of A with respect to \Sigma is | ||
| 72번째 줄: | 74번째 줄: | ||
| − | + | The <math>\lambda</math> - length is <math>\lambda_{\Sigma}(A) : = e^{l_{\Sigma}(A)/2}</math> | |
Note : \lambda_{\Sigma}(A) in \mathbb{R}_{> 0 } | Note : \lambda_{\Sigma}(A) in \mathbb{R}_{> 0 } | ||
2011년 8월 13일 (토) 08:18 판
introduction
review of hyperbolic geometry
- horocycle
- 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
shear coordinate
- E an edge (diagonal), T a triangulation (collection of edges or diagonals)
- \(\Sigma\) a point in the Teichmuller space (metric)
- shear coordinate \(\tau_{\Sigma}(E,T)\)
- for tropical version of shear coordiante, see lamination and shear coordinates on a marked surface
decorated Teichmuller space
\def decorated Teichmuller space \(\tilde{T}(S,M)\) is
- a point in T(S,M) (i.e. hyperbolic metric)
- a choic of horocycle around each cusp from M (i.e. marked point)
\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})
history
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
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
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- http://ncatlab.org/nlab/show/HomePage
experts on the field
links