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Teichmüller 空間英語Teichmüller space

數學中, Teichmüller 空間 of a (real) topological (or differential) 曲面 , is a space that parametrizes 復結構 on up to the action of 同胚s that are isotopic to the identity homeomorphism. 中的每一點可視為「標記」 Riemann 曲面的一個等價類, where a "marking" is an isotopy class of homeomorphisms from to itself.

It can also be viewed as a 模空間 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 for a surface of genus . In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space.

Teichmüller 空間具有一個典範的複流形結構, 以及豐富的自然度量. 考察這些各式各樣結構的幾何特徵是一個富有成果的研究方向.

Teichmüller 空間以 Oswald Teichmüller英語Oswald Teichmüller 的名字命名.

歷史

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Riemann 曲面模空間及相關的 Fuchs 群英語Fuchsian group的研究始自 Bernhard Riemann 的工作. Riemann 明白, 刻畫虧格為的曲面上的各種不同的復結構需要個參數. 從19世紀晚期到20世紀早期的 Teichmüller 空間早期研究是幾何式的, 其基礎是將 Riemann 曲面解釋為雙曲型曲面, 主要貢獻者包括 Felix Klein, Henri Poincaré, Paul Koebe英語Paul Koebe, Jakob Nielsen英語Jakob Nielsen (mathematician), Robert Fricke英語Robert FrickeWerner Fenchel英語Werner Fenchel.

Teichmüller 在模研究方面的主要貢獻是將擬共形映射引入此課題. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by Lars AhlforsLipman Bers英語Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers).

The geometric vein in the study of Teichmüller space was revived following the work of William Thurston in the late seventies, who introduced a geometric compactification which he used in his study of the 映射類群英語mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmüller space, and this is a very active subject of research in 幾何群論.

定義

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Teichmüller space from complex structures

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為一個可定向光滑曲面 (即2維微分流形). 非正式來講, 的 Teichmueller空間 上的 Riemann 曲面結構全體的同倫等價類空間.

其正式定義之一如下. 的兩個復結構 稱為等價的, 如果存在一個可微同胚 使得:

  • 是全純的 (the differential is complex linear at each point, for the structures at the source and at the target) ;
  • 的恆同自映射同倫等價 (存在連續映射 滿足 ).

上的復結構全體在此等價關係下的商空間.

另一種等價的定義是: is the space of pairs where is a Riemann surface and a diffeomorphism, and two pairs are regarded as equivalent if is isotopic to a holomorphic diffeomorphism. Such a pair is called a marked Riemann surface; the marking being the diffeomeorphism; another definition of markings is by systems of curves.[1]

There are two simple examples that are immediately computed from the 單值化定理: there is a unique complex structure on the 球面 (參見 Riemann 球面) and there are two on (the complex plane and the unit disk) and in each case the group of positive diffeomorphisms is 可縮的. Thus the Teichmüller space of is a single point and that of contains exactly two points.

A slightly more involved example is the open annulus, for which the Teichmüller space is the interval (the complex structure associated to is the Riemann surface ).

The Teichmüller space of the torus and flat metrics

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The next example is the 環面 . In this case any complex structure can be realised by a Riemann surface of the form (a 復橢圓曲線) for a complex number where is the complex half-plane

There is then a map , mapping to the pair . It is a bijection[2] and thus the Teichmüller space of is

Identifying with the Euclidean plane each point in Teichmüller space can also be viewed as a marked flat structure on . Thus the Teichmüller space is in bijection with the set of pairs where is a flat surface and is a diffeomorphism up to isotopy on .

有限型曲面

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這是一類其 Teichmüller 空間研究得最多的曲面, 其中也包括了閉曲面. 一個曲面如果與一個去掉有限個點的緊曲面可微同胚, 則稱為是有限型的. 若 虧格閉曲面英語closed surface, 則從 去掉 個點而得的曲面通常記作 , 而其 Teichmüller 空間記作 .

Teichmüller 空間與雙曲度量

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Every finite type orientable surface other than the ones above admits complete Riemannian metrics of constant curvature . For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the 單值化定理. Thus if the Teichmüller space can be realised as the set of marked hyperbolic surfaces of genus with cusps, that is the set of pairs where is an hyperbolic surface and is a diffeomorphism, modulo the equivalence relation where and are identified is is isotopic to an isometry.

Teichmüller 空間的拓撲

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In all cases computed above there is an obvious topology on Teichmüller space. In the general case there are many natural ways to topologise , perhaps the simplest is via hyperbolic metrics and length functions.

If is a 閉曲線 on and a marked hyperbolic surface then one is homotopic to a unique 閉測地線英語closed geodesic on (up to parametrisation). The value at of the length function associated to (the homotopy class of) is then:

Let be the set of 簡單閉曲線s on . Then the map defined by is an embedding. The space has the 積拓撲 and is endowed with the 誘導拓撲. With this topology is homeomorphic to .

In fact one can obtain an embedding with curves,[3] and even .[4] In both case one can use the embedding to give a geometric proof of the homeomorphism above.

其它較為簡單的 Teichmüller 空間舉例

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There is a unique complete hyperbolic metric on the three-holed sphere[5] and so the Teichmüller space is a point (this also follows from the dimension formula of the previous paragraph).

The Teichmüller spaces and are naturally realised as the upper half-plane, as can be seen using Fenchel–Nielsen coordinates.

Teichmüller 空間與共形結構

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Instead of complex structures of hyperbolic metrics one can define Teichmüller space using conformal structures. Indeed, conformal structures are the same as complex structures in two (real) dimensions.[6] Moreover, the Uniformisation Theorem also implies that in each conformal class of Riemannian metrics on a surface there is a unique metric of constant curvature.

Teichmüller 空間作為表示空間

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Yet another interpretation of Teichmüller space is as a representation space for surface groups. If is hyperbolic, of finite type and is the 基本群 of then Teichmüller space is in natural bijection with:

  • The set of injective representations with discrete image, up to conjugation by an element of , if is compact ;
  • In general, the set of such representations, with the added condition that those elements of which are represented by curves freely homotopic to a puncture are sent to parabolic elements of , again up to conjugation by an element of .

The map sends a marked hyperbolic structure to the composition where is the monodromy of the hyperbolic structure and is the isomorphism induced by .

Note that this realises as a closed subset of which endows it with a topology. This can be used to see the homeomorphism directly.[7]

This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group is replaced by an arbitrary semisimple 李群.

關於範疇的註記

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All definitions above can be made in the 拓撲空間範疇 instead of the 微分流形範疇, and this does not change the objects.

無限維 Teichmüller 空間

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Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to ). Another example of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces.[8][9]

Action of the mapping class group and relation to moduli space

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The map to moduli space

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There is a map from Teichmüller space to the 模空間 of Riemann surfaces diffeomorphic to , defined by . It is a covering map, and since is 單連通的 it is the orbifold universal cover for the moduli space.

Action of the mapping class group

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The 映射類群英語mapping class group of is the coset group of the diffeomorphism group of by the normal subgroup of those that are isotopic to the identity (the same definition can be made with homeomorphisms instead of diffeomorphisms and this does not change the resulting group). The group of diffeomorphisms acts naturally on Teichmüller space by

If is a mapping class and two diffeomorphisms representing it then they are isotopic. Thus the classes of and are the same in Teichmüller space, and the action above factorises through the mapping class group.

The action of the mapping class group on the Teichmüller space is properly discontinuous, and the quotient is the moduli space.

不動點

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The Nielsen realisation problem asks whether any finite group of the mapping class group has a global fixed point (a point fixed by all group elements) in Teichmüller space. In more classical terms the question is: can every finite subgroup of be realised as a group of isometries of some complete hyperbolic metric on (or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by Steven Kerckhoff英語Steven Kerckhoff.[10]

坐標

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Fenchel–Nielsen 坐標

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The Fenchel–Nielsen coordinates (so named after Werner Fenchel英語Werner Fenchel and Jakob Nielsen英語Jakob Nielsen (mathematician)) on the Teichmüller space are associated to a pants decomposition of the surface . This is a decomposition of into pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in Teichmüller space, and another real parameter called the twist which is more involved to define.[11]

In case of a closed surface of genus there are curves in a pants decomposition and we get parameters, which is the dimension of . The Fenchel–Nielsen coordinates in fact define a homeomorphism .[12]

In the case of a surface with punctures some pairs of pants are "degenerate" (they have a cusp) and give only two length and twist parameters. Again in this case the Fenchel–Nielsen coordinates define a homeomorphism .

Shear 坐標

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If the surface admits ideal triangulations (whose vertices are exactly the punctures). By the formula for the Euler characteristic such a triangulation has triangles. An hyperbolic structure on determines an (unique up to isotopy) diffeomorphism sending every triangle to an hyperbolic ideal triangle, thus a point in . The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation.[13] There are such parameters which can each take any value in , and the completeness of the structure corresponds to a linear equation and thus we get the right dimension . These coordinates are called shear coordinates.

For closed surfaces, a pair of pants can be decomposed as the union of two ideal triangles (it can be seen as an incomplete hyperbolic metric on the three-holed sphere[14]). Thus we also get shear coordinates on .

Earthquakes

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A simple earthquake path in Teichmüller space is a path determined by varying a single shear or length Fenchel–Nielsen coordinate (for a fixed ideal triangulation of a surface). The name comes from seeing the ideal triangles or the pants as tectonic plates and the shear as plate motion.

More generally one can do earthquakes along geodesic laminations. A theorem of Thurston then states that two points in Teichmüller space are joined by a unique earthquake path.

分析理論

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擬共形映射

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A quasiconformal mapping between two Riemann surfaces is a homeomorphism which deforms the conformal structure in a bounded manner over the surface. More precisely it is differentiable almost everywhere and there is a constant , called the dilatation, such that

where are the derivatives in a conformal coordinate and its conjugate .

There are quasi-conformal mappings in every isotopy class and so an alternative definition for The Teichmüller space is as follows. Fix a Riemann surface diffeomorphic to , and Teichmüller space is in natural bijection with the marked surfaces where is a quasiconformal mapping, up to the same equivalence relation as above.

二次微分與 Bers 嵌入

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Image of the Bers embedding of a punctured torus' 2-dimensional Teichmüller space

With the definition above, if there is a natural map from Teichmüller space to the space of -equivariant solutions to the Beltrami differential equation.[15] These give rise, via the Schwarzian derivative, to quadratic differentials on .[16] The space of those is a complex space of complex dimension , and the image of Teichmüller space is an open set.[17] This map is called the Bers embedding.

A quadratic differential on can be represented by a translation surface conformal to .

Teichmüller 映射

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Teichmüller's theorem[18] states that between two marked Riemann surfaces and there is always a unique quasiconformal mapping in the isotopy class of which has minimal dilatation. This map is called a Teichmüller mapping.

In the geometric picture this means that for every two diffeomorphic Riemann surfaces and diffeomorphism there exists two polygons representing and an affine map sending one to the other, which has smallest dilatation among all quasiconformal maps .

度量

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Teichmüller 度量

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If and the Teichmüller mapping between them has dilatation then the Teichmüller distance between them is by definition . This indeed defines a distance on which induces its topology, and for which it is complete. This is the metric most commonly used for the study of the metric geometry of Teichmüller space. In particular it is of interest to geometric group theorists.

There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on , which is not symmetric.[19]

Weil–Petersson 度量

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Quadratic differentials on a Riemann surface are identified with the tangent space at to Teichmüller space.[20] The Weil–Petersson metric is the Riemannian metric defined by the inner product on quadratic differentials.

緊緻化

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There are several inequivalent compactifications of Teichmüller spaces that have been studied. Several of the earlier compactifications depend on the choice of a point in Teichmüller space so are not invariant under the modular group, which can be inconvenient. William Thurston later found a compactification without this disadvantage, which has become the most widely used compactification.

Thurston 緊緻化

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By looking at the hyperbolic lengths of simple closed curves for each point in Teichmüller space and taking the closure in the (infinite-dimensional) projective space, Thurston (1988) introduced a compactification whose points at infinity correspond to projective measured laminations. The compactified space is homeomorphic to a closed ball. This Thurston compactification is acted on continuously by the modular group. In particular any element of the modular group has a fixed point in Thurston's compactification, which Thurston used in his classification of elements of the modular group.

Bers 緊緻化

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The Bers compactification is given by taking the closure of the image of the Bers embedding of Teichmüller space, studied by Bers (1970). The Bers embedding depends on the choice of a point in Teichmüller space so is not invariant under the modular group, and in fact the modular group does not act continuously on the Bers compactification.

Teichmüller 緊緻化

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The "points at infinity" in the Teichmüller compactification consist of geodesic rays (for the Teichmüller metric) starting at a fixed basepoint. This compactification depends on the choice of basepoint so is not acted on by the modular group, and in fact Kerckhoff showed that the action of the modular group on Teichmüller space does not extend to a continuous action on this compactification.

Gardiner–Masur 緊緻化

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Gardiner & Masur (1991)considered a compactification similar to the Thurston compactification, but using extremal length rather than hyperbolic length. The modular group acts continuously on this compactification, but they showed that their compactification has strictly more points at infinity.

Large-scale geometry

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There has been an extensive study of the geometric properties of Teichmüller space endowed with the Teichmüller metric. Known large-scale properties include:

  • Teichmüller space contains flat subspaces of dimension , and there are no higher-dimensional quasi-isometrically embedded flats.[21]
  • In particular, if or or then is not hyperbolic.

On the other hand, Teichmüller space exhibits several properties characteristic of hyperbolic spaces, such as:

  • Some geodesics behave like they do in hyperbolic space.[22]
  • Random walks on Teichmüller space converge almost surely to a point on the Thurston boundary.[23]

Some of these features can be explained by the study of maps from Teichmüller space to the curve complex, which is known to be hyperbolic.

復幾何

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Bers 嵌入賦予 一個復結構, 使之同構於 的一個開子集.

Metrics coming from the complex structure

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Since Teichmüller space is a complex manifold it carries a Carathéodory 度量英語Carathéodory metric. Teichmüller space is Kobayashi hyperbolic and its 小林昭七度量英語Kobayashi metric coincides with the Teichmüller metric.[24] This latter result is used in Royden's proof that the mapping class group is the full group of isometries for the Teichmüller metric.

The Bers embedding realises Teichmüller space as a 全純域 and hence it also carries a Bergman 度量英語Bergman metric.

Teichmüller 空間上的 Kähler 度量

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The Weil–Petersson metric is Kähler but it is not complete.

鄭紹遠 and 丘成桐 showed that there is a unique complete Kähler–愛因斯坦度量英語Kähler–Einstein metric on Teichmüller space.[25] It has constant negative scalar curvature.

Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by McMullen (2000) that is Kähler-hyperbolic.

Equivalence of metrics

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With the exception of the incomplete Weil–Petersson metric, all metrics on Teichmüller space introduced here are 擬等距同構 to each other.[26]

參見

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引用

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  1. ^ Imayoshi & Taniguchi 1992,第14頁.
  2. ^ Imayoshi & Taniguchi 1992,第13頁.
  3. ^ Imayoshi & Taniguchi 1992,Theorem 3.12.
  4. ^ Hamenstädt, Ursula. Length functions and parameterizations of Teichmüller space for surfaces with cusps. Annales Acad. Scient. Fenn. 2003, 28: 75–88. 
  5. ^ Ratcliffe 2006,Theorem 9.8.8.
  6. ^ Imayoshi & Taniguchi 1992,Theorem 1.7.
  7. ^ Imayoshi & Taniguchi 1992,Theorem 2.25.
  8. ^ Ghys, Etienne. Laminations par surfaces de Riemann. Panor. Synthèses. 1999, 8: 49–95. MR 1760843. 
  9. ^ Deroin, Bertrand. Nonrigidity of hyperbolic surfaces laminations. Proceedings of the American Mathematical Society. 2007, 135 (3): 873–881. MR 2262885. doi:10.1090/s0002-9939-06-08579-0. 
  10. ^ Kerckhoff 1983.
  11. ^ Imayoshi & Taniguchi 1992,第61頁.
  12. ^ Imayoshi & Taniguchi 1992,Theorem 3.10.
  13. ^ Thurston 1988,第40頁.
  14. ^ Thurston 1988,第42頁.
  15. ^ Ahlfors 2006,第69頁.
  16. ^ Ahlfors 2006,第71頁.
  17. ^ Ahlfors 2006,Chapter VI.C.
  18. ^ Ahlfors 2006,第96頁.
  19. ^ Thurston, William, Minimal stretch maps between hyperbolic surfaces, 1998 [1986], Bibcode:1998math......1039T, arXiv:math/9801039可免費查閱 
  20. ^ Ahlfors 2006, Chapter VI.D
  21. ^ Eskin, Alex; Masur, Howard; Rafi, Kasra. Large scale rank of Teichmüller space. Duke Mathematical Journal. 2017, 166 (8): 1517–1572. arXiv:1307.3733可免費查閱. doi:10.1215/00127094-0000006X. 
  22. ^ Rafi, Kasra. Hyperbolicity in Teichmüller space. Geometry & Topology. 2014, 18 (5): 3025–3053. arXiv:1011.6004可免費查閱. doi:10.2140/gt.2014.18.3025. 
  23. ^ Duchin, Moon. Thin triangles and a multiplicative ergodic theorem for Teichmüller geometry (學位論文). University of Chicago. 2005. 
  24. ^ Royden, Halsey L. Report on the Teichmüller metric. Proc. Natl. Acad. Sci. U.S.A. 1970, 65 (3): 497–499. Bibcode:1970PNAS...65..497R. MR 0259115. PMC 282934可免費查閱. PMID 16591819. doi:10.1073/pnas.65.3.497. 
  25. ^ Cheng, Shiu Yuen; Yau, Shing Tung. On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation. Comm. Pure Appl. Math. 1980, 33 (4): 507–544. MR 0575736. doi:10.1002/cpa.3160330404. 
  26. ^ Yeung, Sai-Kee. Quasi-isometry of metrics on Teichmüller spaces. Int. Math. Res. Not. 2005, 2005 (4): 239–255. MR 2128436. doi:10.1155/IMRN.2005.239. 

參考文獻

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擴展閱讀

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