ian short thursday 29 november 2012 - open...
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Riemann surfaces
Ian Short
Thursday 29 November 2012
Complex analysis andgeometry in the plane
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
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
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
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
?
z 7!az + b
cz + d, ad � bc 6= 0
Conformally equivalent domains
z 7!az + b
cz + d, ad � bc 6= 0
Riemann mapping theorem
Holes and punctures
?
Holes and punctures
?
Koebe's theorem
Fixed points, Koebe uniformization and circle packings
Zheng-Xu He and Oded Schramm
The Annals of Mathematics, 1993
Koebe's theorem
Fixed points, Koebe uniformization and circle packings
Zheng-Xu He and Oded Schramm
The Annals of Mathematics, 1993
Koebe's theorem
Fixed points, Koebe uniformization and circle packings
Zheng-Xu He and Oded Schramm
The Annals of Mathematics, 1993
The hyperbolic plane
�(a ; b) = infnR
jdz jy
: path from a to bo
The hyperbolic plane
�(a ; b) = infnR
jdz jy
: path from a to bo
The hyperbolic plane
�(a ; b) = infnR
jdz jy
: path from a to bo
Quasi-isometric graph
Quasi-isometric graph
Quasi-isometric graph
Disc model of the hyperbolic plane
Three geometries
curvature + 0 �
triangle angles > � � < �
parallel lines 0 1 1
circumference 2� sin r 2�r 2� sinh r
Three geometries
curvature + 0 �
triangle angles > � � < �
parallel lines 0 1 1
circumference 2� sin r 2�r 2� sinh r
Three geometries
curvature + 0 �
triangle angles > � � < �
parallel lines 0 1 1
circumference 2� sin r 2�r 2� sinh r
Three geometries
curvature + 0 �
triangle angles > � � < �
parallel lines 0 1 1
circumference 2� sin r 2�r 2� sinh r
Three geometries
curvature + 0 �
triangle angles > � � < �
parallel lines 0 1 1
circumference 2� sin r 2�r 2� sinh r
Surfaces
Surfaces
Surfaces
Surfaces
Classification of compact surfaces
Conway's ZIP proof
George K. Francis & Je�rey R. Weeks
American Mathematical Monthly, 1999
Classification of compact surfaces
Conway's ZIP proof
George K. Francis & Je�rey 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
Conformal : analytic and bijective
Analytic maps on Riemann surfaces
Conformal : analytic and bijective
Analytic maps on Riemann surfaces
Conformal : analytic and bijective
Analytic maps on Riemann surfaces
Conformal : analytic and bijective
Lego Riemann surfaces
Algebraic curves
w7 = 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! � 1zi
The modular surface
G = hz 7! z + 1; z 7! � 1zi
The modular surface
G = hz 7! z + 1; z 7! � 1zi
The uniformisationtheorem
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 .
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 .
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 .
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
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.
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.
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
sinh� sinh� > 1
Geometry of Riemann surfaces
sinh� sinh� > 1
Conformal maps
Pick's version of Schwarz's lemma
�S (f (z ); f (w)) 6 �R(z ;w)
Pick's version of Schwarz's lemma
�S (f (z ); f (w)) 6 �R(z ;w)
Pick's version of Schwarz's lemma
�S (f (z ); f (w)) 6 �R(z ;w)
Group of conformal symmetries
Conformal symmetry group of R = H=G is N (G)=G
Group of conformal symmetries
Conformal symmetry group of R = H=G is N (G)=G
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.
Intersting exercise. Prove from the Riemann�Hurwitz
formula that an analytic map from a compact Riemann surface
(genus > 2) to itself is either constant or a conformal symmetry.
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 Riemann�Hurwitz
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
z 7!az + b
cz + da ; b; c; d 2 C ad � bc = 1
Schwarz reflection principle and Möbius maps
z 7!az + b
cz + da ; b; c; d 2 C ad � bc = 1
Schwarz reflection principle and Möbius maps
z 7!az + b
cz + da ; b; c; d 2 C ad � bc = 1
Schwarz reflection principle and Möbius maps
z 7!az + b
cz + da ; b; c; d 2 C ad � bc = 1
Schwarz reflection principle and Möbius maps
z 7!az + b
cz + da ; b; c; d 2 C ad � bc = 1
Connectivity 3+
Conformal symmetry group is �nite group of Möbius transfor-
mations
Connectivity 3+
Conformal symmetry group is �nite group of Möbius transfor-
mations
Connectivity 3+
Conformal symmetry group is �nite group of Möbius transfor-
mations
Connectivity 3+
Conformal symmetry group is �nite group of Möbius transfor-
mations
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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!