топология / Farb, Margalit, A primer on mapping class groups

In the sequence of four inequalities above, the assumption that distinguishes the x–direction from the y–direction is the assumption that K ≥ 1. Since we now have K = 1, we have attained symmetry between the horizontal and vertical directions.

We must have that all four inequalities in the above calculation are equalities. For the first of the four inequalities, namely, inequal ity (11.3), to be an equality it must be that f takes horizontal line segments of X to horizontal line segments of Y . By symmetry, f must also take vertical line segments of X to vertical line segments of Y . We therefore have

f (x, y) = (u(x), v(y)).

For the second inequality above, namely, inequality (11.2) to be an equality, we must have that the direction of maximal stretch for f is the x–direction at almost every point of X. By symmetry, the direction of maximal stretch for

f must also be the y–direction at almost every point. Thus, |u′(x)| = |v′(y)| at almost every point (x, y) X, and jf (x, y) = u′(x)v′(y) = u′(x)2 = v′(y)2 (we are using the fact that f is orientation-preserving to say that jf ≥ 0) and Kf (x, y) = 1. Since Kf = sup Kf (x, y) = 1, we must have that Kf (x, y) ≡ 1. For the third inequality above, the Cauchy–Schwarz inequality, to be an equality, we must have that

jf (x, y)/Kf (x, y) = jf (x, y) = u′(x)2 = v′(y)2

is constant almost everywhere. For the fourth inequality to be an equality, we must have that jf (x, y) is equal to K = 1 almost everywhere. Thus, u′(x) = v′(x) = 1 almost everywhere, and f (x, y) = (x, y) almost everywhere. Since f is a homeomorphism, it follows that f (x, y) = (x, y)

¨

11.5 PROOF OF TEICHMULLER'S UNIQUENESS THEOREM

The proof of Teichm ¨uller's uniqueness theorem (Theorem 11.7) is almost exactly the solution to Gr¨otzsch's problem just given. The only difficulty is to prove the analogue of the inequality (11.3) with the rectangle X replaced by a closed Riemann surface. To do this we will first need the fo llowing lemma.

In what follows ℓq denotes the Euclidean length of a path with respect to a holomorphic quadratic differential q.

Lemma 11.9 Let qY be a holomorphic quadratic differential on a closed Riemann surface Y . Let h : Y → Y be a homeomorphism that is homotopic to the identity. Then there exists a constant M ≥ 0 with the following property: any arc α : [0, 1] → Y embedded in a leaf of the horizontal foliation for qY satisfies

ℓqY (h(α)) ≥ ℓqY (α) − M.

The point of Lemma 11.9 is that, while the constant M depends on the homeomorphism h, it does not depend on the arc α.

Proof. Let H(x, t) : Y ×[0, 1] → Y be any homotopy from h to the identity. The ℓqY –distance between H(x, 0) and H(x, 1) is a continuous function Y → R. Since Y is compact, it attains a maximum N ≥ 0.

Denote by δ0 and δ1 the arcs H(α(0), 1 − t) and H(α(1), t). Note that ℓqY (δ0) ≤ N and ℓqY (δ1) ≤ N . The concatenation of arcs δ0 h(α) δ1 is homotopic to α, relative to endpoints. Since α is embedded in a horizontal leaf of qY , it minimizes the length of any arc in its relative homotopy class. We thus have

ℓqY (α) ≤ ℓqY (δ0 h(α) δ1)

= ℓqY (δ0) + ℓqY (h(α)) + ℓqY (δ0) ≤ ℓqY (h(α)) + 2N.

We are now ready to prove the analogue for closed Riemann surfaces of inequality (11.3), which estimates the mean horizontal stretching of a map homotopic to a Teichm ¨uller mapping. In the statement of the following proposition, fx denotes the derivative of f in the direction of the horizontal foliation for qX , and Area(q) denotes the Euclidean area of the holomorphic quadratic differential q.

Proposition 11.10 Let h : X → Y be a Teichmuller¨ mapping between closed Riemann surfaces. Suppose that h has initial differential is qX , terminal differential is qY , and dilatation K. Let f : X → Y be any homeomorphism that is homotopic to h, and that is smooth outside a finite number

of points. Then

Z

|fx| dA ≥ K Area(qX ).

X

We note that in the case that each of the horizontal leaves of qX is closed, the proof of Proposition 11.10 is quite similar to the solution of Gr¨otzsch's problem. The case when qX has non-closed leaves is more subtle.

Proof of Proposition 11.10. Consider the function δ : X × R≥0 → R≥0 given by

Z L

δ(p, L) = |fx| dx.

−L

If αp,L is the horizontal arc of length 2L centered at p then δ(p, L) is the integral of |fx| along αp,L. Note that δ is not defined everywhere; specifically, it is not defined at any (p, L) where p lies at a horizontal distance less than L from a zero of qX . This is a set of measure zero in X × R≥0 since there are only finitely many zeros of qX , each with finitely many separatrices in the horizontal foliation for qX .

Where δ(p, L) is defined we have

δ(p, L) = ℓqY (f (αp,L)).

The Teichm ¨uller map h takes αp,L to an arc of qY –length 2KL. It follows from Lemma 11.9 that

ℓqY (f (αp,L)) ≥ 2KL − M

where M is independent of p and of L. Thus

On the other hand, Fubini's theorem gives

Combining the above two equations gives

Z

2L |fx| dA ≥ (2KL − M ) Area(qX ).

X

334 CHAPTER 11

Dviding both sides of this inequality by 2L and allowing L to tend to infinity

As alluded to above, the proof of Teichm ¨uller's uniqueness theorem is now verbatim the proof of Gr¨otzsch's problem, now that we have Proposition 11.10. Thus, we do not repeat the argument here.

¨

11.6 PROOF OF TEICHMULLER'S EXISTENCE THEOREM

The goal of this section is to prove Teichm ¨uller's existence theorem (Theorem 11.6). The key idea here is to reinterpret the existence problem for Teichm ¨uller maps as the problem of proving surjectivity of a natural “exponential map” Ω : QD(X) → Teich(Sg ). We will define such a map, prove continuity and properness, and deduce surjectivity by general topology, namely invariance of domain. We now begin executing this strategy.

Let X be a closed Riemann surface and let q QD(X). Recall that by equation (11.1) the absolute value of q at a point does not depend on the local expression for q. Thus we can define a norm on QD(X) via

Z

kqk = |q|.

X

Let QD1(X) denote the open unit ball in QD(X). For q QD1(X), set

1 + kqk

K = 1 − kqk.

As in Section 11.3, we can construct a Riemann surface Y and a Teichm ¨uller mapping h : X → Y with initial differential q and horizontal stretch factor K. By identifying X with Sg , we can regard X as a point X Teich(Sg ). Then, using h to pull back the complex structure on Y to a complex structure on Sg , we can think of Y as a point Y Teich(Sg ). This procedure therefore defines a map

Ω : QD1(X) → Teich(Sg ).

Teichm ¨uller's existence theorem then amounts to the statement that Ω is surjective. Indeed, let Z be a Riemann surface and let f : X → Z be a homeomorphism. Teichm ¨uller's existence theorem states that there is a Teichm ¨uller map h : X → Z in the homotopy class of f . We can use f to

pull back the complex structure on Z to a complex structure on X = Sg in order to obtain a point Z Teich(Sg ). If there exists q QD1(X) such that Ω(q) = Z, then this exactly means that this there is a Teichm ¨uller map X → Z in the homotopy class of f , as desired.

In order to analyze Ω as a map between topological spaces we must specify a topology on QD1(X). The norm k · k gives QD(X) a topology. This topology is compatible with the topology that QD(X) inherits from the standard topology on R6g−6 via the homeomorphism QD(X) → R6g−6 of Theorem ??. The space QD1(X) then inherits the subspace topology from QD(X).

Proposition 11.11 Let g ≥ 1. The map Ω : QD1(X) → Teich(Sg ) is continuous.

Proposition 11.11 is far from obvious. For example, even if q QD(X) has the property that its horizontal foliation has only closed leaves, there will be a nearby q′ QD(X) with leaves that are not closed. If we stretch along both foliations by a factor of K, there is no simple reason why the resulting points of Teich(Sg ) should be close to each other.

Proposition 11.11 represents the main content of our proof of Teichm ¨uller's existence theorem. We will prove it below as a corollary of the measurable Riemann mapping theorem.

Proposition 11.12 The map Ω : QD1(X) → Teich(Sg ) is proper.

Proof. Let κ : Teich(Sg ) → R be defined by the following formula. For Y Teich(Sg ) we represent Y by a marked Riemann surface Y . Then we set

κ(Y) = inf{Kh | h : X → Y a quasiconformal homeomorphism isotopic to the identity}.

We claim that the map κ is continuous. Indeed given two nearby points Y

and Y′ in Teich(Sg ), we can represent them by nearby elements of DF (π1(Sg ), PSL(2, R)). The (marked) fundamental domains for these representations can be made

K–quasiconformally equivalent for any K > 1 by taking Y′ sufficiently close to Y. Say that Y′ is represented by a marked Riemann surface Y ′. By Teichm ¨uller uniqueness theorem, the infimum κ(Y) is realized by some

336 CHAPTER 11

Kh : X → Y . Since the composition of a Kh–quasiconformal map with a K–quasiconformal map is KhK–quasiconformal it follows that κ(Y′) can be made arbitrarily close to κ(Y) by taking Y′ close to Y.

Let A Teich(Sg ) be compact. Since κ is continuous, κ|A attains a maximum, say M ≥ 0. We claim that Ω−1(A) is contained in the closed ball of radius (M − 1)/(M + 1) about the origin in QD1(X). Since Ω−1(A) is closed by Proposition 11.11, this claim will imply that Ω−1(A) is compact, so that Ω is proper.

We now prove the claim. Let q Ω−1(A). By the definition of Ω, there is a Teichm ¨uller map h : X → Ω(q) that is isotopic to the identity and that has dilatation

1 + kqk

Kh = 1 − kqk.

By Teichm ¨uller's uniqueness theorem (Theorem 11.7), any quasiconformal homeomorphism X → Ω(q) isotopic to the identity must have dilatation at least Kh. It follows then from the definition of M that

Brouwer's invariance of domain theorem [32] states that any injective continuous map Rn → Rn is an open map. We have the following straightforward consequence, which is the final piece needed for our p roof of Teichm ¨uller's existence theorem.

THEOREM 11.13 Any proper injective continuous map Rn → Rn is a homeomorphism.

With the above ingredients in place, we can prove Teichm ¨uller's existence theorem (Theorem 11.6).

Proof of Teichmuller's¨ existence theorem. The map

Ω : QD1(X) → Teich(Sg )

is injective by Teichm ¨uller's uniqueness theorem (Theorem 11.7), proper (Proposition 11.12), and continuous (Proposition 11.11). Also QD1(X) is a real vector space of dimension greater than or equal to 6g−6 (Theorem 11.4) and Teich(Sg ) ≈ R6g−6 (Theorem 10.5). Since QD1(X) contains a subspace homeomorphic to R6g−6, Theorem 11.13 implies that the map Ω is

BELTRAMI DIFFERENTIALS, THE MEASURABLE RIEMANN MAPPING THEO-

REM, AND THE CONTINUITY OF Ω

Fix a closed Riemann surface X of genus g ≥ 2. To prove the continuity of

Ω : QD1(X) → Teich(Sg ) we will factor Ω as

Ω1 ∞ Ω2

Ω : QD1(X) → L (U ) → Teich(Sg )

where U C is the upper half-plane. The image of Ω1 will consist of (equivalence classes of) complex-valued functions that come from Beltrami differentials, which we now define.

Recall that an ellipse field on a Riemann surface X is a choice of ellipse in the each tangent space T Xp at each point p X. An ellipse field is smooth if, when written in local coordinates, it varies smoothly. A quasiconformal homeomorphism f : X → Y determines an ellipse field on X that is defined, and smooth, almost everywhere, as follows. Given a poi nt p X, we take the ellipse in T Xp that is the preimage of the unit circle in T Yf (p) under the derivative of f (when the latter is defined). Since this ellipse is only well-defined up to scale, we always choose the ellipse to have unit area.

In any chart, we can encode an ellipse field by a complex-value d function µ, called the complex dilatation. It is given locally by the formula

µ= fz¯ . fz

Recall from the beginning of the chapter that |µ| < 1 if and only if f is orientation-preserving.

An ellipse field is smooth if and only if the corresponding µ is smooth in

each chart. The dilatation Kf (p) is locally given by the formula

1 + |µ(p)| Kf (p) = 1 − |µ(p)|.

It is also possible to calculate the angle, in any chart, of the direction of maximal stretch of df . It is given by 12 arg(µ). This information completely determines the corresponding ellipse field; namely, the ell ipse at the point p is (up to scale) the unit-area ellipse with major axis having angle 12 arg(µ) and with ratio of the lengths of the axes being (1 + |µ(p)|)/(1 − |µ(p)|).

To make a definition of µ that is independent of charts, we define it as a (−1, 1)–form , which simply means that µ transforms under change of coordinates by the formula

where z and w are two overlapping sets of coordinates. Since (dw/dz)/(dw/dz) lies on the unit circle, the (−1, 1)–form µ gives rise to a well-defined func-

tion |ν| : X → R. We say that the differential µ is a Beltrami differential if |µ| is essentially bounded, that is, off of a set of measure zero it is a bounded function.

We can interpret equation (11.4) geometrically as follows: if a transition map between charts rotates by an angle α at a point then µ is multiplied by ei2α. Since the angle of the ellipse field is locally given by 12 arg(µ), this means that the differential of a change of coordinates map from one chart to another takes the ellipse field corresponding to the first loc al expression for µ to the ellipse field corresponding to the second local expres sion for µ.

As the Riemann surface X is a quotient X = U/π1(X) of the upper halfplane U by conformal automorphisms, we can think of the coefficient µ of a Beltrami differential as a bounded, π1(X)–equivariant, measurable function on U . That is, we can use the interior of some preferred fundamental domain for π1(X) in U as the target of a single chart, take µ as above on that chart, and extend µ to all of U using the action of π1(X) on U and the change of coordinates formula (11.4). By reflecting across the real ax is, i.e., by setting µ(¯z) = µ(z), we can also think of µ as an element of L∞(C). Recall that two functions represent the same point of L∞(C) if they are equal almost everywhere. We give L∞(C) its usual topology of almost everywhere uniform convergence.

We saw above how every quasiconformal homeomorphism f of a Riemann

surface X gives rise to a bounded unit-area ellipse field µf on X. We also saw that any such unit-area ellipse field is equivalent to giv ing a (π1(X)– equivariant) element µ L∞(U ). Which such µ L∞(C) occur? This question gives rise to the fundamental “inverse problem” fo r Beltrami differentials: given any µ L∞(C), is it possible to find a quasiconformal homeomorphism f : C → C so that f satisfies the Beltrami equation

µfz = fz¯

almost everywhere? Very generally, the answer is “yes.”

THEOREM 11.14 (Measurable Riemann mapping theorem) Let µ L∞(C) and suppose kµk∞ ≤ 1. There exists a unique quasiconformal homeomorphism f µ : Cb → Cb that fixes 0, 1, and ∞ and satisfies almost everywhere

the Beltrami equation

µfzµ = fz¯µ.

Further, f µ is smooth wherever µ is, and f µ varies complex analytically with respect to µ.

By the uniqueness statement in Theorem 11.14 we see that if µ(¯z) = µ(z) then f µ restricts to a self-map of the upper half-plane U of C. The uniqueness statement, together with the π1(X)–equivariance of µ, also implies that f µ is π1(X)–equivariant.

There is a long history concerning the existence of solutions to the Beltrami equation. The case where µ is continuous was first proven by Lavrentiev [106], and where µ is measurable by Morrey [130]. The analytic dependence of the solution on µ is due to Ahlfors–Bers [2].

The proof of Theorem 11.14 is beyond the scope of this book. We refer the reader to [2] or [3, Chapter 5] for the proof. Assuming this theorem, we can now prove Proposition 11.11, which states that Ω : QD1(X) → Teich(Sg ) is continuous.

Proof of Proposition 11.11. Let X be a Riemann surface of genus g ≥ 2. We take X to be homeomorphically identified with Sg , and so X represents a point of Teich(Sg ). After defining the maps

Ω1 : QD1(X) → L∞(U ) and Ω2 : L∞(U ) → Teich(Sg )

that we alluded to at the start of this section, we will prove that both Ω1 and Ω2 are continuous. We will then show that Ω = Ω2 ◦ Ω1.

Let q QD(X). Just as we were able to convert a Beltrami differential into a π1(X)–equivariant function U → C, so are we able to convert q into a π1(X)–equivariant function qe : U → C defined almost everywhere on U . What is more, if we fix the covering map U → X and the preferred fundamental domain in U ahead of time, then the map q 7→qe is a welldefined function.

We can then define Ω1 : QD1(X) → L∞(U ) by setting Ω1(0) = 0 and by setting

Ω1(q)(z) = kqk qe(z)/|qe(z)|

for q 6= 0. Here, kqk is the norm of the vector q in the vector space QD(X), and |qe(z)| is the absolute value of the complex number qe(z). Note that if qe transforms by dz2 then q/e |qe| transforms by dz/dz, as desired. Informally,

dz2/(dzdz) = dz/dz.

e ≈

We claim that the map Ω1 is continuous. Indeed, as a function on X U , the element Ω1(q) L∞(U ) is “equivariant” with respect to the π1(X) action on U , in the sense that equation (11.1) is satisfied. Thus, if we ch ange q QD1(X) by a small amount in one chart (say the chart given by the preferred fundamental domain), then by (11.1) the function Ω1(q) changes by a small amount. It also follows from equation (11.1) that kΩ1(q)k∞ = kqk < 1.

The map Ω2 : L∞(U ) → Teich(Sg ) is given by the measurable Riemann mapping theorem. To make this precise, we begin by realizing X as a representation

f : C → C be the function guaranteed by the measurable Riemann mapping theorem (Theorem 11.14) and restrict it to U . If we conjugate each element in the image of ρ by f µ, we obtain a new Riemann surface X′, and f µ induces a homeomorphism X → X′. We can regard X′ as a point of Teich(Sg ). The last sentence in the statement of the measurable Riemann mapping theorem (Theorem 11.14) implies that Ω2 is continuous.

It only remains to check that Ω : QD1(X) → Teich(Sg ) is equal to the composition Ω2 ◦ Ω1. Let q QD1(X) and suppose that at some point