Monodromy According to H.A.Schwarz 1 Riemann Mapping

Monodromy according to H.A.Schwarz

1 Riemann mapping theorem and reﬂection principle

In this subsection we recall some background in the theory of complex analytic functions.

Theorem 1.1 (Riemann, Koebe,1912) Let G be a simply connected domain in C, distinct from C. Let D be the open disc with radius 1 and center 0. Then there exists a bi-analytic map f from G onto D (i.e f is bijective and both f and its inverse are analytic). Furthermore, given z0 ∈ G, there exists a unique such function with the property that 0 f(z0) = 0 and f (z0) > 0.

This theorem was announced by Riemann, but a rigorous proof was given only years later by P.Koebe in 1912. It is possible to extend the function f continuously to the boundary of G if this boundary is suﬃciently well behaved.

Theorem 1.2 Let G be a simply connected domain in C, distinct from C and suppose that its boundary is piecewise smooth. Let D be the open disc with radius 1 and center 0. Then there exists a bijective map f from G onto D, bi-analytic on G and continuous on G (here G denotes the closure of G). In particular, the boundary of G is mapped bijectively onto the boundary of D.

The next theorem concerns analytic continuation across a smooth boundary of a domain.

Theorem 1.3 Let G1,G2 ⊂ C be domains whose boundaries are simply closed continuous curves which have a smooth open curve B in common. Suppose we have two functions fi : Gi → C, for i = 1, 2, analytic on Gi and continuous on Gi ∪ B with the property that f1(z) = f2(z) for all z ∈ B. Then f2 is an analytic continuation of f1 to G2 across B.

An important application of Theorem 1.3 is the following.

Theorem 1.4 (Schwarz’s reflection principle) Let G ⊂ C be a domain whose boundary is a simply closed continuous curve and which contains an open part B which is part of a circle or a straight line. Suppose we have a continuous function G ∪ B → C, analytic on G such that f(B) lies on a circle or straightline. Denote by ∗ the reflections in B,as well as in f(B). Define the function g(z) = f(z∗)∗ on G∗ ∪ B, where G∗ is the reflected image of G with respect to B. Then g(z) is an analytic continuation of f to G∗.

The proof consists of the insight that g(z) is again analytic and that f(z) = g(z) for all z ∈ B. Application of Theorem 1.3 yields the desired result.

1 2 Schwarzian derivative

An important quantity for diﬀerential equations of order 2 is the Schwarzian derivative. Let w be a meromorphic function in some domain in C. Then we deﬁne w00 0 1 w00 2 w000 3 w00 2 S(w) = − = − . w0 2 w0 w0 2 w0 A crucial property is the following. Proposition 2.1 Let w, w˜ be two meromorphic functions on a connected domain G ⊂ C. Then we have a b S(w) = S(w, ˜ z) ⇐⇒ there exists ∈ GL(2, ) such that w˜ = aw+b on G. c d C cw+d A property of the Schwarzian is that it detects points where w is not locally conformal. More precisely,

Proposition 2.2 We consider w(z) near a point z0 and suppose it can be written as a (z − z0) u(z) where u is holomorphic around z0 and u(z0) 6= 0. Then, S(w) has a second order pole in z0 and we have the following behaviours of S(w) near z0,

1. If a = ±1 the Schwarzian S(w) is holomorphic in z0.

2. If a 6= 0, ±1 the Schwarzian S(w) has a second order pole in z0 and a principal part 1−a2 starting with 2 . 2(z−z0)

If w is holomorphic in z0 and w(z0) 6= 0 we replace w by w − w(z0). If w = log(z − z0) + u(z) with u holomorphic in z0, then S(w) has a second order pole in 1 z0 with principal part starting with 2 . 2(z−z0) If w = zau(1/z) where u is a power series in 1/z and u(0) 6= 0, then S(w) is given by a 1−a2 power series in 1/z with lowest order term 2z2 . The proof is by direct computation. The connection with second order diﬀerential equations is given as follows. Proposition 2.3 Consider the second order diﬀerential equation y00 + p(z)y0 + q(z)y = 0 with p, q meromorphic functions in some connected domain. Then, for any two independent local solutions y1, y2 of this equation we have 0 2 S(y1/y2) = 2q − p − p /2.

Proof Direct computation. 2 A direct consequence of the above two propositions is the following. Corollary 2.4 Let p, q be meromorphic functions and Q = 2q − p0 − p2/2. Then the solutions to the non-linear diﬀerential equation S(w) = Q in the unknown function w is 00 0 given by w = y1/y2 where y1, y2 are any two independent solutions of y + py + qy = 0.

2 3 Schwarz’s mapping theorem

First a little geometry.

Deﬁnition 3.1 A curvilinear triangle is a connected open subset of C ∪ ∞ = P1 whose boundary is the union of three open segments of a circle or straight line and three points. The segments are called the edges of the triangles, the points are called the vertices. It is an exercise to prove that, given the vertices and the corresponding angles (< π), a curvilinear triangle exists and is uniquely determined This can be seen best by taking the vertices to be 0, 1, ∞. Then the edges connected to ∞ are actually straight lines. More generally, a curvilinear triangle in C ∪ ∞ = P1 is determined by its angles (in clockwise ordering) up to a M¨obiustransformation. Let z0 be a point in the upper half plane H = {z ∈ C|=(z) > 0} and let f, g be two independent solutions of the hypergeometric equation near z0. The quotient η(z) = f/g, considered as a map from H to P1, is called the Schwarz map and we have the following picture and theorem.

D(1)

D(z)

D(∝)

0 1 ∝ D(0)

Theorem 3.2 (Schwarz) Let a, b, c be real parameters and deﬁne λ = |1 − c|, µ = |c − a − b|, ν = |a − b|. Suppose that 0 ≤ λ, µ, ν < 1. Then the map η(z) = f/g maps H ∪ R one-to-one and conformally onto a curvilinear triangle. The vertices correspond to the points η(0), η(1), η(∞) and the corresponding angles are λπ, µπ, νπ.

The following three ingredients give an indication of why this theorem may hold. – The map η(z) is locally bijective in every point of H. Notice that η0(z) = (f 0g − fg0)/g2. The determinant f 0g − fg0 is the Wronskian determinant of our equation and equals z−c(1 − z)c−a−b−1. In particular it is non-zero in H. When g has a zero at some point z1 we simply consider 1/η(z) instead. Since f and g cannot vanish at the same time in a regular point, we have f(z1) 6= 0 and so 1/η and hence η are conformal in z1. – The map η(z) maps the segments (∞, 0), (0, 1), (1, ∞) to segments of circles or straight lines. For example, since a, b, c ∈ R we have two real solutions on (0, 1),

3 namely f˜ = z1−cF (a−c+1, b−c+1, 2−c|z) andgF ˜ (a, b, c|z). Clearly, the function η˜(z) = f/˜ g˜ maps (0, 1) on a segment of R. Since f, g are C-linear combinations of f,˜ g˜ we see that η(z) is a M¨obiustransform ofη ˜(z). Hence η(z) maps (0, 1) to a segment of a circle or a straight line. We have a similar argument for the segments (1, ∞) and (−∞, 0).

– The map η(z) maps a small neighbourhood of 0 to a sector with angle |1 − c| = λ and similarly for 1, ∞. This follows from the fact that, for example, near z = 0 the functionη ˜(z) is given by z1−c times a power series in z with constant term 1. We argue similarly for the neighbourhoods of 1, ∞.

However, appealing as the above considerations may be, they do not give us an acceptable proof of Theorem 3.2. Here follows a valid proof. Proof[of Theorem 3.2] Let T be a curvilinear triangle with angles λπ, µπ, νπ (in clockwise order). Notice also that the unit disc and the complex upper half plane H are conformally equivalent. Using the Riemann mapping theorem there exists a bicontinuous map w : H → T , analytic on H (it is agreed that w(∞) is the vertex with angle νπ). We continue w analytically to the lower half plane and use the Schwarz reflection principle. In the application of this principle it is important to consider which segment of R is transversed. In the following picture we continue analytically along a loop γ which first crosses (0, 1) and then returns by crossing (1, ∞). On the right of the picture we follow the image of γ and use the reflection principle whenever γ crosses the real line.

D(1)

D(z)

D(∝) 0 1 ∝ D(0)

After continuation of w(z) along γ the continued function will be w(z) followed by two reﬂections. Hence there exist A, B, C, D ∈ C with AD − BC 6= 0 such that the continued Aw(z)+B function is Cw(z)+D . Consider S(w). From the invariance of the Schwarz derivative with respect to M¨obiustransforms it follows that S(w), after continuation along γ returns to the original S(w). The same holds for the simple loop encircling z = 0. Hence S(w) is an analytic function in C minus 0, 1. The behaviour of S(w) near z = 0 follows from Proposition 2.2. It turns out that S(w) has a double pole at z = 0 with a principal part

4 1−λ2 1−µ2 that begins with 2z2 . Similarly, S(w) behaves around z = 1 like 2z2 . Finally, for z near ∞ Proposition 2.2 tells us that S(w) has a power series expansion in 1/z beginning 1−ν2 with 2z2 . Write S(w)/2 = Q. We conclude that Q is a rational function with double poles in 0, 1 and prescribed leading terms in the expansions around 0, 1, ∞. Application of Proposition 2.3 shows that functions w which satisfy S(w)/2 = Q are given by y1/y2 00 where y1, y2 are linearly independent solutions of y + Qy = 0. From the shape of Q we deduce that this is a Fuchsian diﬀerential equation with Riemann scheme

0 1 ∞ (1 + λ)/2 (1 + µ)/2 (−1 + ν)/2 (1 − λ)/2 (1 − µ)/2 (−1 − ν)/2

For example, near z = 0 the indicial equation reads ρ(ρ − 1) + (1 − λ2)/4 = 0, hence the above local exponents follow. Assuming that 1 − c = λ and c − a − b = µ we can multiply the solutions of this equations by z(−1+λ)/2(z − 1)(−1+µ)/2 to get solutions of the equation with Riemann scheme 0 1 ∞ 0 1 ∞ λ µ (1 + ν − λ − µ)/2 which equals 1 − c c − a − b a 0 0 (1 − ν − λ − µ)/2 0 0 b

For all cases 1 − c = ±λ, c − a − b = ±µ we can work this out in a similar way and ﬁnish the proof. 2