Riemann surfaces Ian Short Thursday 29 November 2012
Complex analysis and geometry in the plane
Complex differentiability
Complex differentiability
Complex differentiability
Complex differentiability
Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w )
Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w )
Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w )
Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w ) Conformal : analytic and bijective
Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w ) Conformal : analytic and bijective
Complex differentiability f (z + w ) = f (z ) + f 0 (z )w + "(w ) Conformal : analytic and bijective
Topologically equivalent domains
Topologically equivalent domains
Topologically equivalent domains
Conformally equivalent domains?
Conformally equivalent domains z 7! az + b, ad bc 6= 0 cz + d
Riemann mapping theorem
Holes and punctures?
Holes and punctures?
Koebe's theorem
Koebe's theorem
Koebe's theorem Fixed points, Koebe uniformization and circle packings Zheng-Xu He and Oded Schramm The Annals of Mathematics, 1993
The hyperbolic plane
The hyperbolic plane (a; b) = inf n R jdz j y o : path from a to b
The hyperbolic plane (a; b) = inf n R jdz j y o : path from a to b
Quasi-isometric graph
Quasi-isometric graph
Quasi-isometric graph
Disc model of the hyperbolic plane
Three geometries
Three geometries curvature + 0
Three geometries curvature + 0 triangle angles > <
Three geometries curvature + 0 triangle angles > < parallel lines 0 1 1
Three geometries curvature + 0 triangle angles > < parallel lines 0 1 1 circumference 2 sin r 2r 2 sinh r
Surfaces
Surfaces
Surfaces
Surfaces
Classification of compact surfaces
Classification of compact surfaces Conway's ZIP proof George K. Francis & Jerey R. Weeks American Mathematical Monthly, 1999
Classification of surfaces
Jacob's ladder
The infinite Loch Ness monster
Riemann surfaces
Riemann surfaces
Riemann surfaces
Riemann surfaces
Plane domains
The Riemann sphere
Analytic maps on Riemann surfaces
Analytic maps on Riemann surfaces
Analytic maps on Riemann surfaces
Analytic maps on Riemann surfaces Conformal : analytic and bijective
Lego Riemann surfaces
Algebraic curves w 7 = z 3 2z 2 + z
Quotient spaces
Cylinder G = hz 7! z + 1i
Cylinder G = hz 7! z + 1i
Cylinder G = hz 7! z + 1i
Torus G = hz 7! z + 1; z 7! z + ii
Torus G = hz 7! z + 1; z 7! z + ii
Torus G = hz 7! z + 1; z 7! z + ii
Cylinder G = hz 7! z + 1i
Cylinder G = hz 7! z + 1i
Cylinder G = hz 7! z + 1i
Cylinder G = hz 7! z + 1i
Cylinder G = hz 7! 2z i
Cylinder G = hz 7! 2z i
Cylinder G = hz 7! 2z i
Cylinder G = hz 7! 2z i
The modular surface G = hz 7! z + 1; z 7! 1 z i
The modular surface G = hz 7! z + 1; z 7! 1 z i
The modular surface G = hz 7! z + 1; z 7! 1 z i
The uniformisation theorem
The uniformisation theorem The uniformisation theorem.
The uniformisation theorem The uniformisation theorem. Each Riemann surface is conformally equivalent to =G, where is either S, C, or H,
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.
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.
Spherical Riemann surfaces S=G, where G discrete and has no xed points
Euclidean Riemann surfaces C=G, where G discrete and has no xed points
Hyperbolic Riemann surfaces
Hyperbolic Riemann surfaces
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.
Geometry of Riemann surfaces
Geometry of Riemann surfaces sinh sinh 1
Conformal maps
Pick's version of Schwarz's lemma
Pick's version of Schwarz's lemma S (f (z ); f (w )) R (z ; w )
Pick's version of Schwarz's lemma S (f (z ); f (w )) R (z ; w )
Group of conformal symmetries
Group of conformal symmetries
Group of conformal symmetries Conformal symmetry group of R = H=G is N (G)=G
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.
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.
Conformal symmetries of multiply connected domains
Schwarz reflection principle and Möbius maps
Schwarz reflection principle and Möbius maps
Schwarz reflection principle and Möbius maps
Schwarz reflection principle and Möbius maps
Schwarz reflection principle and Möbius maps z 7! az + b cz + d a; b; c; d 2 C ad bc = 1
Connectivity 3+
Connectivity 3+
Connectivity 3+
Connectivity 3+ Conformal symmetry group is nite group of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite groups of Möbius transformations
Finite orthogonal groups
Finite orthogonal groups
Finite orthogonal groups
Finite orthogonal groups
Finite orthogonal groups
Finite orthogonal groups
Literature
Literature Riemann surfaces, Ahlfors & Sario classic
Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old
Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good
Literature Riemann surfaces, Ahlfors & Sario classic Introduction to Riemann surfaces, Springer old Complex functions, Jones & Singerman good Riemann surfaces, Farkas & Kra hard
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
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
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
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
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.
Thank you!