Riemann Surfaces
                                                           Henry Wilton∗
                                                         Michaelmas 2020
                         Contents
                         1 Analytic and meromorphic functions                                                 3
                             1.1   Analytic functions and their zeroes . . . . . . . . . . . . . . . .        3
                             1.2   Meromorphic functions and singularities . . . . . . . . . . . . .          5
                             1.3   Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . .      8
                             1.4   The complex logarithm . . . . . . . . . . . . . . . . . . . . . . .        9
                         2 Natural boundary, gluing constructions and roots                                 11
                             2.1   Natural boundary . . . . . . . . . . . . . . . . . . . . . . . . . .      11
                             2.2   Agluing construction . . . . . . . . . . . . . . . . . . . . . . . .      13
                             2.3   Complex roots . . . . . . . . . . . . . . . . . . . . . . . . . . . .     15
                         3 Riemann surfaces and analytic maps                                               16
                             3.1   Covering maps . . . . . . . . . . . . . . . . . . . . . . . . . . . .     16
                             3.2   Abstract Riemann surfaces . . . . . . . . . . . . . . . . . . . . .       17
                             3.3   Analytic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . .     20
                         4 Examples of conformal structures and analytic functions                          22
                             4.1   Covering maps and analyticity . . . . . . . . . . . . . . . . . . .       22
                             4.2   Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . .    24
                            ∗Comments and corrections are always welcome.               Please  send  them to
                         h.wilton@maths.cam.ac.uk.
                                                                    1
                           5 Complex tori and the open mapping theorem                                              25
                               5.1    Complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         25
                               5.2    The open mapping theorem . . . . . . . . . . . . . . . . . . . .               27
                               5.3    Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . .           27
                           6 Meromorphic functions and a worked example                                             30
                               6.1    Meromorphic functions . . . . . . . . . . . . . . . . . . . . . . .            30
                               6.2    Aworked example . . . . . . . . . . . . . . . . . . . . . . . . . .            31
                           7 Covering-space theory                                                                  34
                               7.1    Covering-space theory . . . . . . . . . . . . . . . . . . . . . . . .          34
                           8 The monodromy group and the space of germs                                             38
                               8.1    The monodromy group . . . . . . . . . . . . . . . . . . . . . . .              38
                               8.2    The space of germs . . . . . . . . . . . . . . . . . . . . . . . . . .         40
                           9 Uniqueness of analytic continuation and gluing                                         43
                               9.1    Analytic continuation revisited         . . . . . . . . . . . . . . . . . .    43
                               9.2    The classical monodromy theorem . . . . . . . . . . . . . . . .                44
                               9.3    Gluing Riemann surfaces . . . . . . . . . . . . . . . . . . . . . .            45
                           10 More gluing and branching                                                             47
                               10.1 A more detailed gluing example . . . . . . . . . . . . . . . . . .               47
                               10.2 Branching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          50
                           11 The valency and Riemann–Hurwitz theorems                                              52
                               11.1 The valency theorem . . . . . . . . . . . . . . . . . . . . . . . . .            52
                               11.2 Euler characteristic        . . . . . . . . . . . . . . . . . . . . . . . . .    53
                               11.3 The Riemann–Hurwitz theorem . . . . . . . . . . . . . . . . . .                  55
                           12 Applications of Riemann–Hurwitz                                                       56
                               12.1 Immediate consequences . . . . . . . . . . . . . . . . . . . . . .               56
                               12.2 Higher-genus Riemann surfaces . . . . . . . . . . . . . . . . . .                58
                           13 Rational and periodic functions                                                       60
                               13.1 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . .           60
                               13.2 Simply periodic functions . . . . . . . . . . . . . . . . . . . . . .            61
                               13.3 Doubly periodic functions . . . . . . . . . . . . . . . . . . . . . .            63
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                         14 The Weierstrass ℘-function                                                      65
                             14.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    65
                             14.2 Branching properties of ℘Λ . . . . . . . . . . . . . . . . . . . . .       68
                             14.3 An algebraic relation . . . . . . . . . . . . . . . . . . . . . . . . .    69
                         15 More ℘-function and quotients                                                   70
                             15.1 An elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . . . .    70
                             15.2 Classification of elliptic functions . . . . . . . . . . . . . . . . .      71
                             15.3 Quotients of Riemann surfaces . . . . . . . . . . . . . . . . . . .        73
                         16 Uniformisation and its consequences                                             75
                             16.1 The uniformisation theorem . . . . . . . . . . . . . . . . . . . .         75
                             16.2 Classification of Riemann surfaces . . . . . . . . . . . . . . . . .        77
                             16.3 Consequences of uniformisation . . . . . . . . . . . . . . . . . .         80
                         Lecture 1: Analyticandmeromorphicfunctions
                         This course picks up where IB Complex Analysis leaves off. The IB course
                         contains many classical results about complex-differentiable functions of one
                         variable. However, it also raises some natural questions, such as:
                                ‘What do we really mean by multi-valued functions, such as the
                                complex logarithm or mth roots?’
                         The reader may like to keep this question in mind as this course begins. We
                         will see that these questions naturally lead us to define and study a whole new
                         class of mathematical objects, the Riemann surfaces of the course title. As
                         we shall see, Riemann surfaces exhibit a beautiful interplay between analysis
                         and geometry.
                             Since the course leans heavily on some of the results of IB Complex Anal-
                         ysis, we will start by recalling some of the definitions and results from that
                         course.
                         1.1     Analytic functions and their zeroes
                         The functions we study will be defined on domains. A domain is an open,
                         connected subset of the complex plane C. Two of the most important kinds
                                                                    3
                      of domains are the (open) disc
                                            Dˆz ,r∶=˜z ∈CSSz−z S 0 such that f has a power-series expansion
                                    0
                                                   fˆz= ∞ a ˆz−z n
                                                          Q n        0
                           for any z ∈ Dˆz ,r ⊆ D.       n=0
                                          0
                         More precisely, Definition 1.1(i) is the definition of a holomorphic func-
                      tion, and Definition 1.1(ii) is the definition of an analytic function. It is
                      a theorem of IB Complex Analysis that the two definitions coincide. As a
                      consequence, analytic functions are surprisingly rigid: many behaviours that
                      we are used to from real analysis are impossible in the complex setting.
                      Proposition 1.2 (Principle of isolated zeroes). Let f ∶ D → C be an analytic
                      function on a domain D ⊆ C. If fˆz  = 0, then either f is identically zero
                                                          0
                      in a neighbourhood of z , or f is non-zero on a punctured neighbourhood of
                                             0
                      z .
                       0
                      Proof. Unless f ≡ 0 in a neighbourhood of z , there is a minimal m ≥ 0 such
                      that a  ≠0. Hence,                         0
                            m
                                                               m
                                                 fˆz=ˆz−z  gˆz
                                                             0
                      for some analytic function g, defined on an open disc Dˆz ,ρ, with gˆz  ≠ 0.
                                                                             0            0
                      By continuity of g, there is r > 0 with gˆz ≠ 0 for all z ∈ Dˆz ,r, and the
                                                                                   0
                      result follows.
                                                          4