Riemann surfaces. Ian Short. Thursday 29 November 2012
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1 Riemann surfaces Ian Short Thursday 29 November 2012
2 Complex analysis and geometry in the plane
3 Complex differentiability
4 Complex differentiability
5 Complex differentiability
6 Complex differentiability
7 Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w )
8 Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w )
9 Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w )
10 Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w ) Conformal : analytic and bijective
11 Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w ) Conformal : analytic and bijective
12 Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w ) Conformal : analytic and bijective
13 Topologically equivalent domains
14 Topologically equivalent domains
15 Topologically equivalent domains
16 Conformally equivalent domains?
17 Conformally equivalent domains z 7! az + b, ad bc 6= 0 cz + d
18 Riemann mapping theorem
19 Holes and punctures?
20 Holes and punctures?
21 Koebe's theorem
22 Koebe's theorem
23 Koebe's theorem Fixed points, Koebe uniformization and circle packings Zheng-Xu He and Oded Schramm The Annals of Mathematics, 1993
24 The hyperbolic plane
25 The hyperbolic plane (a; b) = inf n R jdz j y o : path from a to b
26 The hyperbolic plane (a; b) = inf n R jdz j y o : path from a to b
27 Quasi-isometric graph
28 Quasi-isometric graph
29 Quasi-isometric graph
30 Disc model of the hyperbolic plane
31 Three geometries
32 Three geometries curvature + 0
33 Three geometries curvature + 0 triangle angles > <
34 Three geometries curvature + 0 triangle angles > < parallel lines 0 1 1
35 Three geometries curvature + 0 triangle angles > < parallel lines circumference 2 sin r 2r 2 sinh r
36 Surfaces
37 Surfaces
38 Surfaces
39 Surfaces
40 Classification of compact surfaces
41 Classification of compact surfaces Conway's ZIP proof George K. Francis & Jerey R. Weeks American Mathematical Monthly, 1999
42 Classification of surfaces
43 Jacob's ladder
44 The infinite Loch Ness monster
45 Riemann surfaces
46 Riemann surfaces
47 Riemann surfaces
48 Riemann surfaces
49 Plane domains
50 The Riemann sphere
51 Analytic maps on Riemann surfaces
52 Analytic maps on Riemann surfaces
53 Analytic maps on Riemann surfaces
54 Analytic maps on Riemann surfaces Conformal : analytic and bijective
55 Lego Riemann surfaces
56 Algebraic curves w 7 = z 3 2z 2 + z
57 Quotient spaces
58 Cylinder G = hz 7! z + 1i
59 Cylinder G = hz 7! z + 1i
60 Cylinder G = hz 7! z + 1i
61 Torus G = hz 7! z + 1; z 7! z + ii
62 Torus G = hz 7! z + 1; z 7! z + ii
63 Torus G = hz 7! z + 1; z 7! z + ii
64 Cylinder G = hz 7! z + 1i
65 Cylinder G = hz 7! z + 1i
66 Cylinder G = hz 7! z + 1i
67 Cylinder G = hz 7! z + 1i
68 Cylinder G = hz 7! 2z i
69 Cylinder G = hz 7! 2z i
70 Cylinder G = hz 7! 2z i
71 Cylinder G = hz 7! 2z i
72 The modular surface G = hz 7! z + 1; z 7! 1 z i
73 The modular surface G = hz 7! z + 1; z 7! 1 z i
74 The modular surface G = hz 7! z + 1; z 7! 1 z i
75 The uniformisation theorem
76 The uniformisation theorem The uniformisation theorem.
77 The uniformisation theorem The uniformisation theorem. Each Riemann surface is conformally equivalent to =G, where is either S, C, or H,
78 The uniformisation theorem The uniformisation theorem. Each Riemann surface is conformally equivalent to =G, where is either S, C, or H, and G is a discrete group of conformal isometries of that acts without xed points.
79 The uniformisation theorem The uniformisation theorem. Each Riemann surface is conformally equivalent to =G, where is either S, C, or H, and G is a discrete group of conformal isometries of that acts without xed points. Further, G is isomorphic to the fundamental group of =G.
80 Spherical Riemann surfaces S=G, where G discrete and has no xed points
81 Euclidean Riemann surfaces C=G, where G discrete and has no xed points
82 Hyperbolic Riemann surfaces
83 Hyperbolic Riemann surfaces
84 Hyperbolic Riemann surfaces Poincaré's theorem. If the sum of each set of equivalent angles is 2, then the group generated by the side pairings is discrete, without xed points, and the polygon is a fundamental region.
85 Geometry of Riemann surfaces
86 Geometry of Riemann surfaces sinh sinh 1
87 Conformal maps
88 Pick's version of Schwarz's lemma
89 Pick's version of Schwarz's lemma S (f (z ); f (w )) R (z ; w )
90 Pick's version of Schwarz's lemma S (f (z ); f (w )) R (z ; w )
91 Group of conformal symmetries
92 Group of conformal symmetries
93 Group of conformal symmetries Conformal symmetry group of R = H=G is N (G)=G
94 Conformal symmetries of compact Riemann surfaces Hurwitz bound. A compact Riemann surface of genus g 2 has no more than 84(g 1) conformal symmetries.
95 Conformal symmetries of compact Riemann surfaces Hurwitz bound. A compact Riemann surface of genus g 2 has no more than 84(g 1) conformal symmetries. Intersting exercise. Prove from the RiemannHurwitz formula that an analytic map from a compact Riemann surface (genus 2) to itself is either constant or a conformal symmetry.
96 Conformal symmetries of multiply connected domains
97 Schwarz reflection principle and Möbius maps
98 Schwarz reflection principle and Möbius maps
99 Schwarz reflection principle and Möbius maps
100 Schwarz reflection principle and Möbius maps
101 Schwarz reflection principle and Möbius maps z 7! az + b cz + d a; b; c; d 2 C ad bc = 1
102 Connectivity 3+
103 Connectivity 3+
104 Connectivity 3+
105 Connectivity 3+ Conformal symmetry group is nite group of Möbius transformations
106 Finite groups of Möbius transformations
107 Finite groups of Möbius transformations
108 Finite groups of Möbius transformations
109 Finite groups of Möbius transformations
110 Finite groups of Möbius transformations
111 Finite groups of Möbius transformations
112 Finite groups of Möbius transformations
113 Finite groups of Möbius transformations
114 Finite groups of Möbius transformations
115 Finite groups of Möbius transformations
116 Finite groups of Möbius transformations
117 Finite groups of Möbius transformations
118 Finite orthogonal groups
119 Finite orthogonal groups
120 Finite orthogonal groups
121 Finite orthogonal groups
122 Finite orthogonal groups
123 Finite orthogonal groups
124 Literature
125 Literature Riemann surfaces, Ahlfors & Sario classic
126 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old
127 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good
128 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good Riemann surfaces, Farkas & Kra hard
129 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good Riemann surfaces, Farkas & Kra hard Lectures on Riemann surfaces, Forster don't know
130 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good Riemann surfaces, Farkas & Kra hard Lectures on Riemann surfaces, Forster don't know Algebraic curves and Riemann surfaces, Miranda good
131 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good Riemann surfaces, Farkas & Kra hard Lectures on Riemann surfaces, Forster don't know Algebraic curves and Riemann surfaces, Miranda good Riemann surfaces, Donaldson new
132 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good Riemann surfaces, Farkas & Kra hard Lectures on Riemann surfaces, Forster don't know Algebraic curves and Riemann surfaces, Miranda good Riemann surfaces, Donaldson new A primer on Riemann surfaces, Beardon good
133 Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good Riemann surfaces, Farkas & Kra hard Lectures on Riemann surfaces, Forster don't know Algebraic curves and Riemann surfaces, Miranda good Riemann surfaces, Donaldson new A primer on Riemann surfaces, Beardon good Acknowledgements Many ideas due to Alan Beardon.
134 Thank you!
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z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1
![z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1](/thumbs/95/123523548.jpg "z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1")
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