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08. Conformal Mapping 1. Conformal (Angle-Preserving) Maps

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08. Conformal Mapping 1. Conformal (Angle-Preserving) Maps (December 1, 2020) 08. Conformal mapping Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/complex/notes 2020-21/08 conformal mapping.pdf] 1. Conformal (angle-preserving) maps 2. Lines and circles and linear fractional transformations 3. Elementary examples 0 4. f (zo) = 0 implies local non-injectivity 5. Automorphisms of the disk and of H 6. Schwarz' lemma 7. Riemann Mapping Theorem 1. Conformal (angle-preserving) maps A complex-valued function f on a non-empty open U ⊂ C is conformal if it preserves angles, in the sense that, for any two smooth parametrized curves γ :[a; b] ! C, δ :[c; d] ! C with γ(a) = zo = δ(c), the angle between γ0(a) and δ0(c) is equal to the angle between (f ◦ γ)0(a) and (f ◦ δ)0(c). The function f is orientation-preserving if the directed angle is preserved. Explicitly, for non-zero z; w 2 C, w z directed angle from z to w = θ 2 , such that = eiθ R jwj jzj That is, w . z eiθ = jwj jzj 0 [1.1] Claim: A holomorphic function f is conformal and orientation-preserving at points zo where f (zo) 6= 0. Proof: This is a direct computation, using the chain rule, noting that f 0 is the complex derivative of f, 0 while γ is the real derivative. With γ(a) = zo = δ(c), 0 0 0 0 0 0 0 0 0 0 (f ◦ γ) (a) = f (γ(a)) · γ (a) = f (zo) · γ (a)(f ◦ δ) (c) = f (δ(a)) · δ (a) = f (zo) · δ (a) so 0 0 0 0 0 0 0 0 (f ◦ γ) (a) . (f ◦ δ) (a) f (zo) γ (a) . f (zo) δ (a) γ (a) . δ (a) 0 0 = 0 0 0 0 = 0 0 j(f ◦ γ) (a)j j(f ◦ δ) (a)j jf (zo) γ (a)j jf (zo) δ (a)j jγ (a)j jδ (a)j showing that directed angles are preserved. === [1.2] Remark: Holomorphic f on U with non-vanishing derivative maps the mutually orthogonal families of lines Re(z) = x = const and Im(z) = y = const to mutually orthogonal curves. 2 For example, letting f(z) = z on the upper half-plane, the lines x ! x + iyo become parabolas 2 2 2 2 x ! x − 2ixyo − yo opening horizontally, and the lines y ! xo + iy become parabolas y ! −y + 2xoy + xo opening vertically. p Letting f(z) = z on C − [0; +1) amounts to looking at inverse images in the previous example, giving mutually orthogonal lines Re(z2) = const and Im(z2) = const, namely, hyperbolas x2 − y2 = const and hyperbolas xy = const. 1 Paul Garrett: 08. Conformal mapping (December 1, 2020) 2. Lines and circles and linear fractional transformations [2.1] Theorem: The collection of lines and circles in C [ f1g is stabilized by linear fractional transformations, and is acted upon transitively by them. a b Proof: Clearly affine maps (z) = (az + b)=d, with ad 6= 0, which are combinations of translations, 0 d rotations, and dilations, preserve lines and circles. Given this, a group-theoretic result greatly simplifies things: [2.2] Claim: (Bruhat decomposition) Let P be the upper-triangular matrices in GL2(C), and the Weyl 0 1 element w = . Then o 1 0 GL2(C) = P t P woP (disjoint union) a b Proof: A group element g = is in P if c = 0, so consider c 6= 0. Then c d a b 1 − d 0 1 ∗ ∗ c = 2 P c d 0 1 1 0 0 ∗ That is, for g 62 P , gP wo \ P 6= φ. That is, gP \ P wo 6= φ, so gP \ P woP 6= φ, and g 2 P woP . === Invoking the Bruhat decomposition, since g 2 P preserves lines and circles, it suffices to prove that the Weyl element wo preserves lines and circles. The real and imaginary parts of wo(z) = 1=z are easily observed: 1 z = z jzj2 so x y Re(1=z) = Im(1=z) = − jzj2 jzj2 2 It is convenient that wo gives the identity map, so the images and inverse images of sets under wo are the same things. Given a line ax + by = c with real a; b; c, the image of the line is x y a − b = c jzj2 jzj2 or ax − by = cx2 + cy2 giving the circle a b a 2 b 2 (x − )2 + (y + )2 = + c c c c The image of a circle jz − zoj = r is 1 − z = r z o or j1 − zo · zj = r · jzj or 2 2 2 2 1 − 2 Re(zo · z) + jzoj · jzj = r · jzj In the special case that jzoj = r, this is a line. Otherwise, it is a circle. 2 Paul Garrett: 08. Conformal mapping (December 1, 2020) Three points on a circle determine the circle completely. A line in C can be viewed determined by two points on it in C, and inevitably passing through 1. The group of linear fractional transformation actions is triply transitive in the sense that it can map any triple of distinct points to any other, so is transitive on circles-and-lines. === a b [2.3] Remark: Beware: except for affine maps (z) = (az + b)=d, with ad 6= 0, linear fractional 0 d transformations do not respect centers of circles. 3. Elementary examples A sector is an open set of the form fz = reiθ : r > 0; a < θ < b; with jb − aj < 2πg Two sectors of the special form fz = reiθ : r > 0; 0 < θ < b; with b < 2πg fz = reiθ : r > 0; 0 < θ < b0; with b0 < 2πg can be mapped holomorphically to each other by a power map 0 0 b b ·log z z −! z b = e b with a continuous logarithm defined on the first sector by iπ Z z=e dw log z = iπ + 1 w integrating along a straight line segment. In polar coordinates, this is simply 0 0 iθ b iθ b re −! r b e b Exhibiting the map as a holomorphic map shows that it preserves angles. Sectors with edges elsewhere than the positive real axis can be rotated, by map z ! µ · z with jµj = 1, to put either edge on the positive real axis. Thus, the problem of mapping one sector to another reduces to that simpler case, by pre-composing and post-composing with rotations: fz = reiθ : r > 0; a < θ < b; with jb − aj < 2πg can be mapped to another sector fz = reiθ : r > 0; a0 < θ < b0; with jb0 − a0j < 2πg by b0−a0 b0−a0 z z b−a 0 z b−a z −! −! −! eia · eia eia eia Again, there is an unambiguous choice of continuous αth power on such a sector, by zα = eα·log z 3 Paul Garrett: 08. Conformal mapping (December 1, 2020) where log z is defined on C with any ray freiθo : r > 0g removed, with this ray not lying in the given sector. Again, such a logarithm can be defined by i(θ +π) Z z=e o dw log z = i(θo + π) + 1 w i integrating along a straight line segment from 1 to z=e (θo +π). That is, all sectors are conformally equivalent. In other words, while z ! zα with real α 6= 0 is conformal at every point other than z = 0, it alters angles at 0 by multiplying angles by α. A bigon is an open subset of CP1 bounded by two arcs, by which we for present purposes we mean either straight line segments, possibly infinite in one or both directions, and/or arcs of circles with angle measure < 2π. 1 The non-degenerate cases are specified by two vertices z1; z2 2 PC , and two distinct circles-or-lines passing through both z1; z2. This configuration cuts PC into four connected components, each of which is a bigon. In the degenerate case that the two points are identical, CP1 is cut into only three connected components. [3.1] Claim: All bigons in the non-degenerate case are conformally equivalent, and are conformally equivalent to the upper half-plane. Proof: Given vertices z1 6= z2, map z1 ! 0 and z2 ! 1 by a linear fractional transformation. Since linear fractional transformations preserve lines and circles, the boundary of the image consists of two straight lines from 0 to 1, so the image is a sector. Rotate the sector until is is of the form freiθ : r > 0; 0 < θ < b; with b < 2πg, and the apply a power map to obtain the upper half-plane H. Thus, any non-degenerate bigon is conformally equivalent to the upper half-plane, so they are equivalent to each other. === [3.2] Claim: All degenerate bigons are conformally equivalent, and are conformally equivalent to the strip 0 < Re(z) < 1. Proof: Map the single vertex to 1. The two bounded arcs, mapped to arcs passing through 1, must become straight lines with no other intersections, thus, parallel. Translate, rotate and dilate to obtain the lines Re(z) = 0 and Re(z) = 1. === [3.3] Remark: An open disk can be considered as a non-degenerate bigon in many ways, as can a half-plane. The Cayley map z + i z −! iz + 1 maps i ! 1, −i ! 0, and 1 ! 1, so maps the unit circle to the real line, because linear fractional transformations preserve circles-and-lines. Thus, it maps the connected components of the complement of the circle to the connected components of the complement of the real line: since 0 ! i, the interior of the circle is mapped to the upper half-plane.
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