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00:02:17 1 History
00:04:46 2 Definitions
00:04:56 2.1 Teichmüller space from complex structures
00:11:18 2.2 The Teichmüller space of the torus and flat metrics
00:15:41 2.3 Finite type surfaces
00:17:06 2.4 Teichmüller spaces and hyperbolic metrics
00:19:52 2.5 The topology on Teichmüller space
00:25:04 2.6 More examples of small Teichmüller spaces
00:26:24 2.7 Teichmüller space and conformal structures
00:27:07 2.8 Teichmüller spaces as representation spaces
00:33:09 2.9 A remark on categories
00:33:32 2.10 Infinite-dimensional Teichmüller spaces
00:34:17 3 Action of the mapping class group and relation to moduli space
00:34:31 3.1 The map to moduli space
00:35:26 3.2 Action of the mapping class group
00:38:32 3.3 Fixed points
00:39:36 4 Coordinates
00:39:45 4.1 Fenchel–Nielsen coordinates
00:43:50 4.2 Shear coordinates
00:47:38 4.3 Earthquakes
00:48:27 5 Analytic theory
00:48:37 5.1 Quasiconformal mappings
00:51:22 5.2 Quadratic differentials and the Bers embedding
00:53:07 5.3 Teichmüller mappings
00:55:11 6 Metrics
00:55:21 6.1 The Teichmüller metric
00:57:14 6.2 The Weil–Petersson metric
00:58:06 7 Compactifications
00:58:48 7.1 Thurston compactification
00:59:46 7.2 Bers compactification
01:00:24 7.3 Teichmüller compactification
01:01:07 7.4 Gardiner–Masur compactification
01:01:36 8 Large-scale geometry
01:04:23 9 Complex geometry
01:05:05 9.1 Metrics coming from the complex structure
01:05:54 9.2 Kähler metrics on Teichmüller space
01:06:37 9.3 Equivalence of metrics
01:06:59 10 See also
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SUMMARY
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In mathematics, the Teichmüller space
T
(
S
)
{\displaystyle T(S)}
of a (real) topological (or differential) surface
S
{\displaystyle S}
, is a space that parametrizes complex structures on
S
{\displaystyle S}
up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in
T
(
S
)
{\displaystyle T(S)}
may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from
S
{\displaystyle S}
to itself.
It can also be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension
6
g
−
6
{\displaystyle 6g-6}
for a surface of genus
g
≥
2
{\displaystyle g\geq 2}
. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space.
The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is a very rich subject of research.
Teichmüller spaces are named after Oswald Teichmüller.
