WOMP 2019: Complex Analysis Minh-Tam Trinh Note. I am giving the second lecture; Meg Doucette is giving the first. 1. Functions Holomorphic functions have a very idiosyncratic geometry. I will explain how some major theorems illustrate this. Henceforth, let C be an open domain, D an open disk, and @D the   positively-oriented loop formed by the boundary of D. Let f C be holomorphic. The residue formula shows that if f =f has no zeros/poles on W ! 0 @D, then Z 1 f 0 X f X (1.1) dz resz. 0 / ordz.f /: 2i f D f D @D f 0 zeros z of f poles z of f (Above, we used the fact that any zero of f of order n corresponds to a simple pole of f 0=f of residue n.) At the same time, the substitution w f .z/ shows that D 1 Z f .z/ 1 Z dw (1.2) 0 dz winding number of f .@D/ around 0: 2i @D f .z/ D 2i f .@D/ w D In short, the number of zeros of f in D, counting multiplicities, is given by a winding number! This result is known as Cauchy’s argument principle. The following result says that if you perturb a function by a very small amount in a fixed neighborhood, then its zeros can move around, but their total multiplicity is conserved. Theorem 1.1 (Rouché). If h C is holomorphic and h < f on @D, then f h has the W ! j j j j C same number of zeros as f in D. Proof. Let ft .z/ f .z/ th.z/. The winding number of ft can’t change discontinuously as t runs D C through the interval Œ0; 1. Let’s use Rouché to prove the fundamental theorem of algebra. We claim that if n n 1 (1.3) f z an 1z a1z a0; D C C C C P i then f has n zeros in C. Indeed, in the statement of Rouché, let h.z/ ai z and pick a D 0 i n 1 disk D so big that zn > h.z/ on the boundary @D. Ä Ä j j j j Corollary 1.2 (Open Mapping). Any nonconstant holomorphic function is an open map. Proof. Using Rouché, show that if f .z/ 0, then D.0/ f .Dı .z// for some ı > 0 and D  > 0. Corollary 1.3 (Maximum Modulus). If f C is nonconstant, then f cannot attain a W ! j j maximum on . Proof. By the open mapping theorem, any z can be perturbed to a nearby value w such 2 2 that f .w/ > f .z/ . j j j j Complex Analysis 2 2. Spaces Let’s use the open mapping theorem to give a second proof of the fundamental theorem of algebra. This proof will involve a new space, namely, the Riemann sphere (2.1) C C : O D [ f1g Let f .z/ be a polynomial. As a map C C, it extends to a map C C that fixes . Since C is ! O ! O 1 O compact and Hausdorff, the image of fO is a closed set. But by the open mapping theorem, it is also open. Since C is connected, we deduce that f .C/ C, whence f .C/ C. O O D O D Theorem 2.1. Every connected, simply-connected Riemann surface is conformally equivalent to one of the following: (1) The plane C. (2) The Riemann sphere C. O (3) The open unit disk D. These options are conformally distinct. We can learn a lot of mathematics simply by computing the conformal automorphism groups of C; C; D. Let’s do them in that order. O Theorem 2.2. Aut.C/ is the group of affine transformations z az b. 7! C Proof. Conformal automorphisms are isometries, so for C, decompose into the composition of a translation and a homothety (i.e., rotation + dilation). a b Theorem 2.3. Aut.C/ PGL2.C/, where the projective equivalence class of GL2.C/ O ' D c d 2 acts by the fractional linear transformation az b (2.2) z C D cz d C on C. O Proof. A map C C is the same thing as a meromorphic function on C. The latter is determined by O ! O O its restriction to C. But we know that meromorphic functions on C are rational functions, so take the form f .z/=g.z/ for some polynomials f; g. Invertibility forces deg f deg g 1. D D Example 2.4. The Cayley transform is the fractional linear transformation given by ! 1 i i z (2.3) z : 1 i D i z C It restricts to a conformal equivalence from the open upper half-plane H to the disk D that sends i 0. What’s the inverse? 7! Before we compute Aut.D/, we prepare some notation. For all ˛ D, the Blaschke factor of ˛ is 2 the fractional linear transformation ! 1 ˛ ˛ z (2.4) ˛.z/ z : D ˛ 1 D 1 ˛z N N One can check directly that ˛ restricts to an involution of D that swaps 0 and ˛. Complex Analysis 3 Theorem 2.5. Every automorphism f Aut.D/ takes the form 2 ! ! u 1 ˛ (2.5) f .z/ u ˛.z/ z D D 1 ˛ 1 N for some u S 1 @D and ˛ D. 2 D 2 Lemma 2.6 (Schwarz). If holomorphic f D D satisfies f .0/ 0, then f .z/ z for all z. W ! D j j Ä j j If moreover equality holds for some nonzero value of z, then f is a rotation. 1 Proof of the theorem. Let ˛ f .0/ and g f ˛. Then Schwarz’s lemma applies to both g 1 D D ı and g , giving 1 (2.6) g.z/ z and g .z/ z j j Ä j j j j Ä j j for all z D. We deduce that g.z/ z g.z/ , whence g.z/ z , for all z. Now the second 2 j j Ä j j Ä j j j j D j j part of the lemma says that g is a rotation. Corollary 2.7. Aut.H/ PSL2.R/, where the action is by fractional linear transformations. ' Proof. Since the Cayley transform is an equivalence H D, it induces an isomorphism Aut.H/ ! ' Aut.D/. Now check what it does at the level of fractional linear transformations. Corollary 2.8. There is a diffeomorphism H PSL2.R/=PSO.2/. ' a b Proof. The stabilizer of i in SL2.R/ consists of the matrices of the form b a . This is precisely SO.2/ SL2.R/.  Another Proof. Under the Cayley transform, the stabilizer of i H corresponds to the stabilizer of 2 0 D. The latter is the rotation group S 1 Aut.D/, which corresponds to SO.2/ Aut.H/. 2   Corollary 2.9. 1.SL2.R// Z. ' 3. Metrics There is a stronger version of Schwarz’s lemma due to Georg Pick. To state it, define the hyperbolic metric on D by 1 ˇ z w ˇÁ (3.1) d.z; w/ tanh ˇ ˇ : D ˇ1 zw ˇ N (Recall that tanh 1.x/ 1 log 1 x , a monotonic function.) To give you some intuition for d, I’ll D 2 1Cx draw the geodesics and horocycles in D. It is similarly easy to draw the pictures for H. Lemma 3.1 (Schwarz–Pick). Any holomorphic f D D is a contraction in the hyperbolic metric: W ! d.f .z/; f .w// d.z; w/. Equivalently, Ä ˇ ˇ ˇ f .z/ f .w/ ˇ ˇ z w ˇ (3.2) ˇ ˇ ˇ ˇ : ˇ1 f .z/f .w/ˇ Ä ˇ1 zw ˇ N Moreover, f is an isometry for d if and only if f is an automorphism. The elements of PSL2.R/ can be classified according to what they do to the hyperbolic geometry of H. For convenience, I’ll work with SL2.R/ instead in what follows. If SL2.R/, then we say 2 that: Complex Analysis 4 (1) Elliptic when tr < 2. Equivalently, is of finite order and its eigenvalues are distinct j j complex conjugates. (2) Parabolic when tr 2. Equivalently, is unipotent. j j D (3) Hyperbolic when tr > 2. Equivalently, can be conjugated into the diagonal torus T of j j ˙ SL2.R/. To be more precise about (3): If is hyperbolic with positive eigenvalues, then we can write ! 1 a (3.3) ˇ ˇ T D 1=a 2 for some ˇ SL2.R/ and a > 0. 2 Let T be the image of T in PSL .R/. The torus T acts on PSL .R/=PSO.2/ by left multiplication. N 2 N 2 Since T R and PSL2.R/=PSO.2/ H, this defines a flow on H. It turns out to be precisely the N ' ' geodesic flow..