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

Riemann surface is simply to change the expression for q on the big chart to be any α dz2 for α C. In this case we get other holomorphic quadratic differentials which are qualitatively different; for instance, some have closed leaves and some do not.

One can construct other examples of holomorphic quadratic differentials using the same idea as above, that is, by gluing together Euclidean polygons by Euclidean translations. For starters, the reader might like to consider the example indicated in Figure 11.5.

As with measured foliations, the polygon construction for holomorphic quadratic differentials has a converse: every holomorphic quadratic differential can be realized in this way. Indeed, the natural coordinates tell us how to cut up the Riemann surface into (finitely many) rectangles, each folia ted by horizontal lines. By placing these rectangles in the Euclidean plane so that the foliations are horizontal, and recording the side identificat ions, we obtain a polygonal description of the surface where the holomorphic quadratic differential is given by dz2. For details, see [160, §11].

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Figure 11.5 An example of a surface obtained by gluing Euclidean rectangles by Euclidean translations. As with the Swiss cross example, this surface can be given the structure of a Riemann surface with a holomorphic quadratic differential.

Quadratic differentials via branched covers. Another way to construct holomorphic quadratic differentials is via branched covers. A map p : X → Y between closed Riemann surfaces is a branched covering map if for any point x X, there are local coordinates where p is given by z 7→zk for some k ≥ 0. Note that if p : X → Y is an orbifold covering in the sense of Chapter 7 or is a branched cover of topological surfaces as above, then we can pull back any complex structure on Y to a complex structure on X, and the map p will then be a branched covering map of Riemann surfaces. The point x is a ramification point of the branched covering if and only if k > 0, and in this case k is called the degree of ramification .

Let p : X → Y be a branched covering map, and suppose we have a holomorphic quadratic differential q on Y . We can lift q to a holomorphic quadratic differential qe on X as follows. If U is some open neighborhood of a point x X, and if there are charts U → C and p(U ) → C where in these coordinates p is given by z 7→ψ(z), then by equation (11.1) we have

x. This agrees with our discussion about branched covers and measured foliations: the foliations for and qe have singularities with (m + 2) prongs and k(m + 2) prongs, respectively.

Of course, in order to lift a holomorphic quadratic differential on a Riemann surface Y , we first need to know how to find a holomorphic quadratic differential on Y . When Y has genus 1 we already proved above that the space of holomorphic quadratic differentials on Y is in bijective correspondence with C. We also explained how to construct branched covers of higher genus topological surfaces over the torus (Figure 11.3), and, again, these give rise to branched covers of Riemann surfaces over Y .

THE VECTOR SPACE OF HOLOMORPHIC QUADRATIC DIFFERENTIALS

Let X be any Riemann surface. It is easy to check that the sum of two holomorphic quadratic differentials on X is a holomorphic quadratic differential on X, as is any complex multiple of a holomorphic quadratic differential. It follows that the set of all holomorphic quadratic differentials on X forms a complex vector space, denoted QD(X).

Our first goal is to give a lower bound on the dimension QD(X). Choose some finite set of points P X. Let KP (X) denote the complex vector space of meromorphic functions f : X → C where f only has simple poles, each occurring at point of P . The following theorem is a special case of Riemann's inequality (see [59, §8.3]).

THEOREM 11.3 Let X be a closed Riemann surface of genus g and let P X be a finite set of points. We have

dimC(KP (X)) ≥ |P | + 1 − g.

We now obtain the desired bound on the dimension of QD(X).

Proposition 11.4 Let X be a closed Riemann surface of genus g. We have

dimC(QD(X)) ≥ 3g − 3.

Proof. Let q0 be an element of QD(X) with only simple zeros. Recall that the horizontal foliation for q0 has three prongs at each singularity. By the Euler–Poincar´e formula (Proposition 11.2), q0 has exactly 4g − 4 zeros.

Let P be the set of 4g − 4 zeros of q0. By Theorem 11.3 we have

dimC(KP (X)) ≥ 3g − 3.

We claim that there is a map QD(X) → KP (X) given by q 7→q/q0. Indeed, the ratio q/q0 is a well-defined function on X by equation (11.1), and the poles of q/q0 are precisely the zeros of q0 at which q does not have

The inequality of Theorem 11.5 turns out to be an equality for g = 2 (in the case g = 1 we have already shown that QD(X) ≈ C). This equality can be deduced from the Riemann–Roch theorem, a deep theorem which sharpens Riemann's theorem. On the other hand, in Section 11.3 we will define a map Ω from the open unit ball in QD(X) to Teich(Sg ). It follows from the definition of Ω and Teichm ¨uller's uniqueness theorem (Theorem 11.7 below) that Ω is injective. By Brouwer's invariance of domain theorem (see Theorem 11.13 below), we then obtain the following theorem.

THEOREM 11.5 Let X be a closed Riemann surface of genus g. We have

dimC(QD(X)) = 3g − 3.

In what follows, we will only need Proposition 11.4, and not Theorem 11.5.

A dimension count for QD(X). For g ≥ 2 we can give a heuristic dimension count for QD(X) that is in the same spirit as our dimension counts for Teich(Sg ) in Chapter 10.

Fix a closed Riemann surface X of genus g ≥ 2, and let q be a holomorphic quadratic differential on X. As discussed above, it is possible to realize X by a Euclidean polygon so that q(z) = dz2 in the “interior chart” of the polygon. We can thus count the dimension of QD(X) by counting the dimension of the space of polygons that give X.

Specifically, we consider connected Euclidean polygons P with the following properties:

·If we identify pairs of parallel sides of P , we obtain a closed surface S of genus g.

·Every vertex of P maps to a point on S with total Eulidean angle 3π.

As in the discussion above, P induces a complex structure on S and the quadratic differential dz2 induces a holomorphic quadratic differential q on S. The second condition on P means that each point of S coming from a vertex of P is a simple zero of q (of the form q(z) = z dz2), and there are no other zeros.

Examples of such polygons exist for every genus; see Figure 11.5 for one example in genus two. The set of these polygons has codimension zero in the space of all polygons giving holomorphic quadratic differentials on S, and so we aim to count the dimension of the space of such polygons P .

Let P and q be as above. By the Euler–Poincar´e formula, q has 4g − 4 simple zeros. Since a simple zero of q accounts for a total interior angle of 3π in P , and since every vertex of P corresponds to a simple zero of q, we see that the sum of the interior angles of P must be

3π(4g − 4) = (12g − 12)π.

The sum of the interior angles of a Euclidean n–gon is (n − 2)π, and so we see that P must have 12g − 10 sides. If we think of each side of P as a vector, we get 2(12g − 10) dimensions' worth of freedom. Since side lengths and angles must match in pairs, we are down to 12g−10 dimensions.

The last pair of sides is determined by the others, and so we lose two more dimensions, giving us 12g − 12, exactly twice what we want.

But as we change the polygon, we are also changing the complex structure on S. We are trying to compute QD(X) for a fixed Riemann surface X. In order to take this into account we must subtract the dimension of the space of all complex structures on S, namely the dimension of Teich(S), which is 6g − 6. We have thus given a heuristic that shows that there are 6g − 6 dimensions worth of possible holomorphic quadratic differentials q on any fixed Riemann surface X.

Dimension count for QD(X): polygons

+2(12g − 10) : Choose 12g − 10 vectors for the polygon's sides.

−(12g − 10) : Sides must match in pairs.

−2 : The last pair of sides is determined by the others.

−(6g − 6) : Subtract the dimension of Teich(S).

= 6g − 6 Total dimensions

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11.3 TEICHMULLER MAPS AND TEICHMULLER'S THEOREMS

We are now ready to describe the homeomorphisms that minimize the quasiconformal dilatation in a given homotopy class, thus giving a solution to Teichm ¨uller's extremal problem.

Let X and Y be two closed Riemann surfaces of genus g. We say that a homeomorphism f : X → Y is a Teichmuller¨ mapping if there are holomorphic quadratic differentials qX and qY on X and Y , respectively, and a positive real number K so that the following two conditions hold:

1.The homeomorphism f takes the zeros of qX to the zeros of qY .

2.If p X is not a zero of qX , then with respect to a set of natural coordinates for qX and for qY based at p and f (p), the homeomorphism f can be written as

f (x + iy) = √Kx + i √1 y. K

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We can concisely describe f by saying that it has initial differential qX , terminal differential qY , and horizontal stretch factor K. (Because of the ambiguity that the horizontal stretch factors K and 1/K give rise to Teichm ¨uller mappings with the same dilatation, we need to keep the distinction between horizontal stretch factor and dilatation.)

Note that the existence of a Teichm ¨uller mapping presupposes that the initial and terminal differentials have the same Euclidean area; this is not a strong assumption, as any holomorphic quadratic differential can be scaled by a real number so as to have unit area, and this rescaling does not change the corresponding horizontal or vertical foliations.

A Teichm ¨uller mapping is not differentiable at the zeros of the initial differential, but it is smooth at all other points. This is why in our definition of quasiconformal homeomorphism we chose to consider homeomorphisms that are smooth outside of a finite number of points, instead o f only considering smooth homeomorphisms.

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Figure 11.6 A Teichm¨uller mapping of the Swiss cross surface.

As a first example of a Teichm ¨uller mapping, consider the hom eomorphism between the Swiss cross Riemann surface and the stretched Swiss cross Riemann surface indicated in Figure 11.6. We can get other, more complicated Teichm ¨uller mappings by rotating the foliation in Figure 11.6 so that it is not parallel to any sides of the polygon, or even so that its slope is irrational.

Given two arbitrary complex structures on a topological surface, it is certainly not obvious that one can construct a Teichm ¨uller mapping taking one structure to the other. However, the next theorem states that these mappings do indeed exist, and moreover they exist in every homotopy class.

THEOREM 11.6 (Teichmuller's¨ existence theorem) Let X and Y be closed Riemann surfaces of genus g ≥ 1 and let f : X → Y be a homeomorphism. There exists a Teichmuller¨ mapping h : X → Y homotopic to f .

The main reason that Teichm ¨uller mappings are so useful and important is that they provide a complete solution to Teichm ¨uller's extremal problem for quasiconformal mappings.

THEOREM 11.7 (Teichmuller's¨ uniqueness theorem) Let h : X → Y be a Teichmuller¨ map between two closed Riemann surfaces of genus g ≥ 1. If f : X → Y is a quasiconformal homeomorphism homotopic to h then

Kf ≥ Kh.

Equality holds if and only if f ◦ h−1 is conformal. In particular, if g ≥ 2, then equality holds if and only if f = h.

The second statement follows from the first statement plus th e fact that the only homotopically trivial conformal homeomorphism of a closed Riemann surface of genus g ≥ 2 is the identity (cf. Proposition 7.7). For a closed Riemann surface X of genus g = 1, the group of conformal automorphisms of X is isomorphic to the group T 2.

We will prove both Theorem 11.6 and Theorem 11.7 later in this chapter.

A minimization theorem for 1–manifolds. The analog of Teichm ¨uller's uniqueness theorem for 1–manifolds is nothing other than th e mean value theorem. Identify S1 with R/Z. Consider the set of all smooth homeomorphisms S1 → S1 that fix 0. Define the dilatation of a smooth homeomorphism f by sup |fx|. By the mean value theorem, the log of the dilatation is minimized in a given homotopy class precisely when f is linear.

GENERATING TEICHMULLER¨ MAPS

In our definition of the Teichm ¨uller map, we were handed two R iemann surfaces and a homeomorphism between them. It is natural to ask if, given an arbitrary closed Riemann surface X, an initial holomorphic quadratic differential qX , and some K > 1, it is always possible to find a Riemann surface Y and a terminal holomorphic quadratic differential qY so that there is a Teichm ¨uller mapping f : X → Y with initial differential qX , terminal differential qY , and horizontal stretch factor K. It turns out that this is always possible.

We now give the construction of the required Teichm ¨uller mapping f for the given input is X, qX , and K. Let X′ be the complement in X of the zeroes of qX . We will refer to the topological surfaces underlying X and X′ as S and S′, respectively. The surface X′ is still a Riemann surface; its complex structure is given by a sufficiently large set of natu ral coordinates with respect to qX , now thought of as a holomorphic quadratic differential on X′. If we compose each chart with the affine map

f (x + iy) = √Kx + i √1 y, K

we obtain a new set of charts on S′, and this new set of charts defines a new complex structure on S′. Call the resulting Riemann surface Y ′. In order to obtain the desired closed Riemann surface Y , we need only note that, by the removable singularity theorem (see, e.g. [44, Theorem V.1.2]) the complex structure Y ′ on S′ extends uniquely to a complex structure Y on all of S.

There is an induced homeomorphism f : X → Y , and an induced holomorphic quadratic differential qY on Y . By construction, f is a Teichm ¨uller mapping with the desired properties. If we fix X and qX , but vary K in (0, ∞), we obtain a 1–parameter family of Riemann surfaces homeomo r- phic to X. Since each of these Riemann surfaces comes with an identific a- tion with X, we can think of this one parameter family as a set of points in Teich(S), where S is the topological surface underlying X. The resulting subset of Teich(S) is called a Teichmuller¨ line. The point X corresponds to K = 1. When we define the metric on Teichm ¨uller space, we will see t hat Teichm ¨uller lines are in fact geodesics.

Since the initial differential qX on X specifies a unique ray in Teich(S), we see that we can think of qX as giving a tangent direction and the pair (qX , K) as giving a tangent vector to Teich(S) at X. Actually, below we will give a norm on QD(X). The resulting map (qX , kqX k) → Teich(S) can be

thought of as an exponential map TX (Teich(S)) → Teich(S). We remark that QD(X) is usually identified with the cotangent space of Teich(S) at the point X.

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11.4 GROTZSCH'S PROBLEM

In 1928, Gr¨otzsch proved the following precursor to the Teichm ¨uller uniqueness theorem. It solves the quasiconformal extremal problem for rectangles. The proof of Teichm ¨uller's uniqueness theorem is based on the solution to Gr¨otzsch's problem.

THEOREM 11.8 (Grotzsch's¨ problem) Let X be the rectangle [0, a]×[0, 1] in R2 and let Y be the rectangle [0, Ka] × [0, 1] for some K ≥ 1. If f : X → Y is any orientation-preserving homeomorphism that is smooth away from a finite number of points, that takes horizontal sid es to horizontal sides, and that takes vertical sides to vertical sides, then

Kf ≥ K

with equality if and only if f is affine.

Note that Theorem 11.8 really gives a statement about general rectangles, as any rectangle is conformally equivalent to one with vertical side length 1.

Figure 11.7 The setup for Gr¨otzsch's problem.

Proof. Let f : X → Y be as in the statement of the theorem. Let Kf (x, y) and jf (x, y) denote the dilatation and the Jacobian of f at the point (x, y)

X.

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which is obtained from the inequality R0a |fx(x, y)| dx ≥ Ka by integrating from 0 to 1 with respect to y.

We are now ready to show that Kf ≥ K. Without loss of generality, assume K ≥ 1. Then

≤Kf (x, y) jf (x, y) dA

X

Z Z

≤ jf (x, y) dA Kf (x, y) dA

X X

≤ (K Area(X))(Kf Area(X)).

The first three inequalities follow from (11.3), (11.2), and the Cauchy– Schwarz inequality. The fourth inequality follows from the fact that Kf (x, y) ≤ Kf for all (x, y) X. It follows that Kf ≥ K.

The lower bound Kf = K is achieved when f is the affine map

A : (x, y) 7→(Kx, y).

It remains to prove the uniqueness statement.

Let f : X → Y be an orientation-preserving homeomorphism as in the statement of the theorem, and assume Kf = K. By replacing f with A−1 ◦f we can assume that K = 1, hence Kf = 1. Our goal now is to show that f is the identity.