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Teichmüller 空间

数学中, Teichmüller 空间 $T(S)$ of a (real) topological (or differential) 曲面 $S$, is a space that parametrizes 复结构 on $S$ up to the action of 同胚s that are isotopic to the identity homeomorphism. $T(S)$ 中的每一点可视为“标记” Riemann 曲面的一个等价类, where a "marking" is an isotopy class of homeomorphisms from $S$ to itself.

It can also be viewed as a 模空间 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 $6g-6$ for a surface of genus $g\geq 2$. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space.

Teichmüller 空间具有一个典范的复流形结构, 以及丰富的自然度量. 考察这些各式各样结构的几何特征是一个富有成果的研究方向.

Teichmüller 空间以 Oswald Teichmüller 的名字命名.

历史

Riemann 曲面的模空间及相关的 Fuchs 群的研究始自 Bernhard Riemann 的工作. Riemann 明白, 刻画亏格为$g\geq 2$的曲面上的各种不同的复结构需要$6g-6$个参数. 从19世纪晚期到20世纪早期的 Teichmüller 空间早期研究是几何式的, 其基础是将 Riemann 曲面解释为双曲型曲面, 主要贡献者包括 Felix Klein, Henri Poincaré, Paul Koebe, Jakob Nielsen, Robert Fricke 及 Werner Fenchel.

Teichmüller 在模研究方面的主要贡献是将拟共形映射引入此课题. 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 Ahlfors 和 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 映射类群 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 几何群论.

定义

Teichmüller space from complex structures

设 $S$ 为一个可定向光滑曲面 (即2维微分流形). 非正式来讲, $S$ 的 Teichmueller空间 $T(S)$ 为 $S$ 上的 Riemann 曲面结构全体的同伦等价类空间.

其正式定义之一如下. $S$ 的两个复结构 $X,Y$ 称为等价的, 如果存在一个可微同胚 $f\in \mathrm {Diff} (S)$ 使得:

• $f$ 是全纯的 (the differential is complex linear at each point, for the structures $X$ at the source and $Y$ at the target) ;
• $f$ 与 $S$ 的恒同自映射同伦等价 (存在连续映射 $\gamma :[0,1]\to \mathrm {Diff} (S)$ 满足 $\gamma (0)=f,\gamma (1)=\mathrm {Id}$).

则 $T(S)$ 为 $S$ 上的复结构全体在此等价关系下的商空间.

另一种等价的定义是: $T(S)$ is the space of pairs $(X,g)$ where $X$ is a Riemann surface and $g:S\to X$ a diffeomorphism, and two pairs $(X,g),(Y,h)$ are regarded as equivalent if $h\circ g^{-1}:X\to Y$ 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.

There are two simple examples that are immediately computed from the 单值化定理: there is a unique complex structure on the 球面 $\mathbb {S} ^{2}$ (参见 Riemann 球面) and there are two on $\mathbb {R} ^{2}$ (the complex plane and the unit disk) and in each case the group of positive diffeomorphisms is 可缩的. Thus the Teichmüller space of $\mathbb {S} ^{2}$ is a single point and that of $\mathbb {R} ^{2}$ contains exactly two points.

A slightly more involved example is the open annulus, for which the Teichmüller space is the interval $[0,1)$ (the complex structure associated to $\lambda$ is the Riemann surface $\{z\in \mathbb {C} :\lambda <|z|<\lambda ^{-1}\}$).

The Teichmüller space of the torus and flat metrics

The next example is the 环面 $\mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}$. In this case any complex structure can be realised by a Riemann surface of the form $\mathbb {C} /(\mathbb {Z} +\tau \mathbb {Z} )$ (a 复椭圆曲线) for a complex number $\tau \in \mathbb {H}$ where $\mathbb {H}$ is the complex half-plane

$\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}.$

There is then a map $\mathbb {H} \to T(\mathbb {T} ^{2})$, mapping $\tau$ to the pair $(\mathbb {C} /(\mathbb {Z} +\tau \mathbb {Z} ),(x,y)\mapsto x+\tau y)$. It is a bijection and thus the Teichmüller space of $\mathbb {T} ^{2}$ is $\mathbb {H} .$

Identifying $\mathbb {C}$ with the Euclidean plane each point in Teichmüller space can also be viewed as a marked flat structure on $T^{2}$. Thus the Teichmüller space is in bijection with the set of pairs $(M,f)$ where $M$ is a flat surface and $f:\mathbb {T} ^{2}\to M$ is a diffeomorphism up to isotopy on $f$.

有限型曲面

这是一类其 Teichmüller 空间研究得最多的曲面, 其中也包括了闭曲面. 一个曲面如果与一个去掉有限个点的紧曲面可微同胚, 则称为是有限型的. 若 $S$ 为亏格为 $g$ 的闭曲面, 则从 $S$ 去掉 $k$ 个点而得的曲面通常记作 $S_{g,k}$, 而其 Teichmüller 空间记作 $T_{g,k}$.

Teichmüller 空间与双曲度量

Every finite type orientable surface other than the ones above admits complete Riemannian metrics of constant curvature $-1$. For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the 单值化定理. Thus if $2g-2+k>0$ the Teichmüller space $T_{g,k}$ can be realised as the set of marked hyperbolic surfaces of genus $g$ with $k$ cusps, that is the set of pairs $(M,f)$ where $M$ is an hyperbolic surface and $f:S\to M$ is a diffeomorphism, modulo the equivalence relation where $(M,f)$ and $(N,g)$ are identified is $f\circ g^{-1}$ is isotopic to an isometry.

Teichmüller 空间的拓扑

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

If $\alpha$ is a 闭曲线 on $S$ and $x=(M,f)$ a marked hyperbolic surface then one $f_{*}\alpha$ is homotopic to a unique 闭测地线 $\alpha _{x}$ on $M$ (up to parametrisation). The value at $x$ of the length function associated to (the homotopy class of) $\alpha$ is then:

$\ell _{\alpha }(x)=\operatorname {Length} (\alpha _{x}).$

Let ${\mathcal {S}}$ be the set of 简单闭曲线s on $S$. Then the map $T(S)\to \mathbb {R} ^{\mathcal {S}}$ defined by $x\mapsto (\ell _{\alpha }(x))_{\alpha \in {\mathcal {S}}}$ is an embedding. The space $\mathbb {R} ^{\mathcal {S}}$ has the 积拓扑 and $T(S)$ is endowed with the 诱导拓扑. With this topology $T(S_{g,k})$ is homeomorphic to $\mathbb {R} ^{6g-6+2k}$.

In fact one can obtain an embedding with $9g-9$ curves, and even $6g-5+2k$. In both case one can use the embedding to give a geometric proof of the homeomorphism above.

其它较为简单的 Teichmüller 空间举例

There is a unique complete hyperbolic metric on the three-holed sphere and so the Teichmüller space $T(S_{0,3})$ is a point (this also follows from the dimension formula of the previous paragraph).

The Teichmüller spaces $T(S_{0,4})$ and $T(S_{1,1})$ are naturally realised as the upper half-plane, as can be seen using Fenchel–Nielsen coordinates.

Teichmüller 空间与共形结构

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. 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 空间作为表示空间

Yet another interpretation of Teichmüller space is as a representation space for surface groups. If $S$ is hyperbolic, of finite type and $\Gamma =\pi _{1}(S)$ is the 基本群 of $S$ then Teichmüller space is in natural bijection with:

• The set of injective representations $\Gamma \to \mathrm {PSL} _{2}(\mathbb {R} )$ with discrete image, up to conjugation by an element of $\mathrm {PSL} _{2}(\mathbb {R} )$, if $S$ is compact ;
• In general, the set of such representations, with the added condition that those elements of $\Gamma$ which are represented by curves freely homotopic to a puncture are sent to parabolic elements of $\mathrm {PSL} _{2}(\mathbb {R} )$, again up to conjugation by an element of $\mathrm {PSL} _{2}(\mathbb {R} )$.

The map sends a marked hyperbolic structure $(M,f)$ to the composition $\rho \circ f_{*}$ where $\rho :\pi _{1}(M)\to \mathrm {PSL} _{2}(\mathbb {R} )$ is the monodromy of the hyperbolic structure and $f_{*}:\pi _{1}(S)\to \pi _{1}(M)$ is the isomorphism induced by $f$.

Note that this realises $T(S)$ as a closed subset of $\mathrm {PSL} _{2}(\mathbb {R} )^{2g+k-1}$ which endows it with a topology. This can be used to see the homeomorphism $T(S)\cong \mathbb {R} ^{6g-6+2k}$ directly.

This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group $\mathrm {PSL} _{2}(\mathbb {R} )$ is replaced by an arbitrary semisimple 李群.

关于范畴的注记

All definitions above can be made in the 拓扑空间范畴 instead of the 微分流形范畴, and this does not change the objects.

无限维 Teichmüller 空间

Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to $\mathbb {R} ^{\mathbb {N} }$). Another example of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces.

Action of the mapping class group and relation to moduli space

The map to moduli space

There is a map from Teichmüller space to the 模空间 of Riemann surfaces diffeomorphic to $S$, defined by $(X,f)\mapsto X$. It is a covering map, and since $T(S)$ is 单连通的 it is the orbifold universal cover for the moduli space.

Action of the mapping class group

The 映射类群 of $S$ is the coset group $MCG(S)$ of the diffeomorphism group of $S$ 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

$g\cdot (X,f)\mapsto (X,f\circ g^{-1}).$

If $\gamma \in MCG(S)$ is a mapping class and $g,h$ two diffeomorphisms representing it then they are isotopic. Thus the classes of $(X,f\circ g^{-1})$ and $(X,f\circ h^{-1})$ are the same in Teichmüller space, and the action above factorises through the mapping class group.

The action of the mapping class group $MCG(S)$ on the Teichmüller space is properly discontinuous, and the quotient is the moduli space.

不动点

主条目：Nielsen realization problem

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 $MCG(S)$ be realised as a group of isometries of some complete hyperbolic metric on $S$ (or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by Steven Kerckhoff.

坐标

Fenchel–Nielsen 坐标

主条目：Fenchel–Nielsen 坐标

The Fenchel–Nielsen coordinates (so named after Werner Fenchel and Jakob Nielsen) on the Teichmüller space $T(S)$ are associated to a pants decomposition of the surface $S$. This is a decomposition of $S$ 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.

In case of a closed surface of genus $g$ there are $3g-3$ curves in a pants decomposition and we get $6g-6$ parameters, which is the dimension of $T(S_{g})$. The Fenchel–Nielsen coordinates in fact define a homeomorphism $T(S_{g})\to ]0,+\infty [^{3g-3}\times \mathbb {R} ^{3g-3}$.

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 $T(S_{g,k})\to ]0,+\infty [^{3g-3+k}\times \mathbb {R} ^{3g-3+k}$.

Shear 坐标

If $k>0$ the surface $S=S_{g,k}$ admits ideal triangulations (whose vertices are exactly the punctures). By the formula for the Euler characteristic such a triangulation has $4g-4+2k$ triangles. An hyperbolic structure $M$ on $S$ determines an (unique up to isotopy) diffeomorphism $S\to M$ sending every triangle to an hyperbolic ideal triangle, thus a point in $T(S)$. The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation. There are $6g-6+3k$ such parameters which can each take any value in $\mathbb {R}$, and the completeness of the structure corresponds to a linear equation and thus we get the right dimension $6g-6+2k$. 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). Thus we also get $3g-3$ shear coordinates on $T(S_{g})$.

Earthquakes

主条目：Earthquake map

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.

分析理论

拟共形映射

主条目：拟共形映射

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 $K\geq 1$, called the dilatation, such that

${\frac {|f_{z}|+|f_{\bar {z}}|}{|f_{z}|-|f_{\bar {z}}|}}\leq K$

where $f_{z},f_{\bar {z}}$ are the derivatives in a conformal coordinate $z$ and its conjugate ${\bar {z}}$.

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 $X$ diffeomorphic to $S$, and Teichmüller space is in natural bijection with the marked surfaces $(Y,g)$ where $g:X\to Y$ is a quasiconformal mapping, up to the same equivalence relation as above.

二次微分与 Bers 嵌入

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

主条目：Schwarz 导数和Bers slice

With the definition above, if $X=\Gamma \setminus \mathbb {H} ^{2}$ there is a natural map from Teichmüller space to the space of $\Gamma$-equivariant solutions to the Beltrami differential equation. These give rise, via the Schwarzian derivative, to quadratic differentials on $X$. The space of those is a complex space of complex dimension $3g-3$, and the image of Teichmüller space is an open set. This map is called the Bers embedding.

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

Teichmüller 映射

Teichmüller's theorem states that between two marked Riemann surfaces $(X,g)$ and $(Y,h)$ there is always a unique quasiconformal mapping $X\to Y$ in the isotopy class of $h\circ g^{-1}$ which has minimal dilatation. This map is called a Teichmüller mapping.

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

度量

Teichmüller 度量

If $x,y\in T(S)$ and the Teichmüller mapping between them has dilatation $K$ then the Teichmüller distance between them is by definition ${\frac {1}{2}}\log K$. This indeed defines a distance on $T(S)$ 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 $T(S)\times T(S)$, which is not symmetric.

Weil–Petersson 度量

主条目：Weil–Petersson 度量

Quadratic differentials on a Riemann surface $X$ are identified with the tangent space at $(X,f)$ to Teichmüller space. The Weil–Petersson metric is the Riemannian metric defined by the $L^{2}$ inner product on quadratic differentials.

紧致化

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 紧致化

主条目：Thurston boundary

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 紧致化

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 紧致化

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 紧致化

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

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 $T(S_{g,k})$ contains flat subspaces of dimension $3g-3+k$, and there are no higher-dimensional quasi-isometrically embedded flats.
• In particular, if $g>1$ or $g=1,k>1$ or $g=0,k>4$ then $T(S_{g,k})$ 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.
• Random walks on Teichmüller space converge almost surely to a point on the Thurston boundary.

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.

复几何

Bers 嵌入赋予 $T(S)$ 一个复结构, 使之同构于 $\mathbb {C} ^{3g-3}$ 的一个开子集.

Metrics coming from the complex structure

Since Teichmüller space is a complex manifold it carries a Carathéodory 度量. Teichmüller space is Kobayashi hyperbolic and its 小林昭七度量 coincides with the Teichmüller metric. 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 全纯域 and hence it also carries a Bergman 度量.

Teichmüller 空间上的 Kähler 度量

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

郑绍远 and 丘成桐 showed that there is a unique complete Kähler–爱因斯坦度量 on Teichmüller space. 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

With the exception of the incomplete Weil–Petersson metric, all metrics on Teichmüller space introduced here are 拟等距同构 to each other.

参见

• 代数曲线的模空间
• p-进Teichmüller理论
• 宇宙际 Teichmüller 理论

参考文献

• Ahlfors, Lars V. Lectures on quasiconformal mappings. Second edition. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard.. American Math. Soc. 2006: viii+162. ISBN 978-0-8218-3644-6. 
• Bers, Lipman, On boundaries of Teichmüller spaces and on Kleinian groups. I, Annals of Mathematics, Second Series, 1970, 91 (3): 570–600, JSTOR 1970638, MR 0297992, doi:10.2307/1970638 
• Fathi, Albert; Laudenbach, François; Poenaru, Valentin. Thurston's work on surfaces. Princeton University Press. 2012: xvi+254. ISBN 978-0-691-14735-2. MR 3053012. 
• Gardiner, Frederic P.; Masur, Howard, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl., 1991, 16 (2–3): 209–237, MR 1099913, doi:10.1080/17476939108814480 
• Imayoshi, Yôichi; Taniguchi, Masahiko. An introduction to Teichmüller spaces. Springer. 1992: xiv+279. ISBN 978-4-431-70088-3. 
• Kerckhoff, Steven P. The Nielsen realization problem. Annals of Mathematics. Second Series. 1983, 117 (2): 235–265. CiteSeerX 10.1.1.353.3593 . JSTOR 2007076. MR 0690845. doi:10.2307/2007076. 
• McMullen, Curtis T., The moduli space of Riemann surfaces is Kähler hyperbolic, Annals of Mathematics, Second Series, 2000, 151 (1): 327–357, JSTOR 121120, MR 1745010, arXiv:math/0010022 , doi:10.2307/121120 
• Ratcliffe, John. Foundations of hyperbolic manifolds, Second edition. Springer. 2006: xii+779. ISBN 978-0387-33197-3. 
• Thurston, William P., On the geometry and dynamics of diffeomorphisms of surfaces, American Mathematical Society. Bulletin. New Series, 1988, 19 (2): 417–431, MR 0956596, doi:10.1090/S0273-0979-1988-15685-6 

扩展阅读

• Bers, Lipman, Finite-dimensional Teichmüller spaces and generalizations, American Mathematical Society. Bulletin. New Series, 1981, 5 (2): 131–172, MR 0621883, doi:10.1090/S0273-0979-1981-14933-8 
• Gardiner, Frederick P., Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), New York: John Wiley & Sons, 1987, ISBN 978-0-471-84539-3, MR 0903027 
• Hubbard, John Hamal, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006, ISBN 978-0-9715766-2-9, MR 2245223 
• Papadopoulos, Athanase (编), Handbook of Teichmüller theory. Vols. I-V, IRMA Lectures in Mathematics and Theoretical Physics, 11, 13, 17, 19, 26, European Mathematical Society (EMS), Zürich, 2007–2016, ISBN 978-3-03719-029-6, MR 2284826, doi:10.4171/029  The last volume contains translations of several of Teichmüller's papers.
• Teichmüller, Oswald, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl., 1939, 1939 (22): 197, JFM 66.1252.01, MR 0003242 
• Teichmüller, Oswald, Ahlfors, Lars V.; Gehring, Frederick W. , 编, Gesammelte Abhandlungen, Berlin, New York: Springer-Verlag, 1982, ISBN 978-3-540-10899-3, MR 0649778 
• Voitsekhovskii, M.I., T/t092330, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4