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A useful rule of thumb in complex analysis is that holomorphic functions behave like large degree polynomials
. This can be evidenced for instance at a “local” level by the Taylor series expansion for a complex analytic function in the disk, or at a “global” level by factorisation theorems such as the Weierstrass factorisation theorem (or the closely related Hadamard factorisation theorem). One can truncate these theorems in a variety of ways (e.g., Taylor’s theorem with remainder) to be able to approximate a holomorphic function by a polynomial on various domains.
In some cases it can be convenient instead to work with polynomials of another variable
such as
(or more generally
for a scaling parameter
). In the case of the Riemann zeta function, defined by meromorphic continuation of the formula
one ends up having the following heuristic approximation in the neighbourhood of a point on the critical line:
Heuristic 1 (Polynomial approximation) Let
be a height, let
be a “typical” element of
, and let
be an integer. Let
be the linear change of variables
for
and some polynomial
of degree
.
The requirement is necessary since the right-hand side is periodic with period
in the
variable (or period
in the
variable), whereas the zeta function is not expected to have any such periodicity, even approximately.
Let us give two non-rigorous justifications of this heuristic. Firstly, it is standard that inside the critical strip (with ) we have an approximate form
of (11). If we group the integers from
to
into
bins depending on what powers of
they lie between, we thus have
For with
and
we heuristically have
and so
where are the partial Dirichlet series
This gives the desired polynomial approximation.
A second non-rigorous justification is as follows. From factorisation theorems such as the Hadamard factorisation theorem we expect to have
where runs over the non-trivial zeroes of
, and there are some additional factors arising from the trivial zeroes and poles of
which we will ignore here; we will also completely ignore the issue of how to renormalise the product to make it converge properly. In the region
, the dominant contribution to this product (besides multiplicative constants) should arise from zeroes
that are also in this region. The Riemann-von Mangoldt formula suggests that for “typical”
one should have about
such zeroes. If one lets
be any enumeration of
zeroes closest to
, and then repeats this set of zeroes periodically by period
, one then expects to have an approximation of the form
again ignoring all issues of convergence. If one writes and
, then Euler’s famous product formula for sine basically gives
(here we are glossing over some technical issues regarding renormalisation of the infinite products, which can be dealt with by studying the asymptotics as ) and hence we expect
This again gives the desired polynomial approximation.
Below the fold we give a rigorous version of the second argument suitable for “microscale” analysis. More precisely, we will show
Theorem 2 Let
be an integer going sufficiently slowly to infinity. Let
go to zero sufficiently slowly depending on
. Let
be drawn uniformly at random from
. Then with probability
(in the limit
), and possibly after adjusting
by
, there exists a polynomial
of degree
and obeying the functional equation (9) below, such that
whenever
.
It should be possible to refine the arguments to extend this theorem to the mesoscale setting by letting be anything growing like
, and
anything growing like
; also we should be able to delete the need to adjust
by
. We have not attempted these optimisations here.
Many conjectures and arguments involving the Riemann zeta function can be heuristically translated into arguments involving the polynomials , which one can view as random degree
polynomials if
is interpreted as a random variable drawn uniformly at random from
. These can be viewed as providing a “toy model” for the theory of the Riemann zeta function, in which the complex analysis is simplified to the study of the zeroes and coefficients of this random polynomial (for instance, the role of the gamma function is now played by a monomial in
). This model also makes the zeta function theory more closely resemble the function field analogues of this theory (in which the analogue of the zeta function is also a polynomial (or a rational function) in some variable
, as per the Weil conjectures). The parameter
is at our disposal to choose, and reflects the scale
at which one wishes to study the zeta function. For “macroscopic” questions, at which one wishes to understand the zeta function at unit scales, it is natural to take
(or very slightly larger), while for “microscopic” questions one would take
close to
and only growing very slowly with
. For the intermediate “mesoscopic” scales one would take
somewhere between
and
. Unfortunately, the statistical properties of
are only understood well at a conjectural level at present; even if one assumes the Riemann hypothesis, our understanding of
is largely restricted to the computation of low moments (e.g., the second or fourth moments) of various linear statistics of
and related functions (e.g.,
,
, or
).
Let’s now heuristically explore the polynomial analogues of this theory in a bit more detail. The Riemann hypothesis basically corresponds to the assertion that all the zeroes of the polynomial
lie on the unit circle
(which, after the change of variables
, corresponds to
being real); in a similar vein, the GUE hypothesis corresponds to
having the asymptotic law of a random scalar
times the characteristic polynomial of a random unitary
matrix. Next, we consider what happens to the functional equation
where
A routine calculation involving Stirling’s formula reveals that
with ; one also has the closely related approximation
when . Since
, applying (5) with
and using the approximation (2) suggests a functional equation for
:
where is the polynomial
with all the coefficients replaced by their complex conjugate. Thus if we write
then the functional equation can be written as
We remark that if we use the heuristic (3) (interpreting the cutoffs in the summation in a suitably vague fashion) then this equation can be viewed as an instance of the Poisson summation formula.
Another consequence of the functional equation is that the zeroes of are symmetric with respect to inversion
across the unit circle. This is of course consistent with the Riemann hypothesis, but does not obviously imply it. The phase
is of little consequence in this functional equation; one could easily conceal it by working with the phase rotation
of
instead.
One consequence of the functional equation is that is real for any
; the same is then true for the derivative
. Among other things, this implies that
cannot vanish unless
does also; thus the zeroes of
will not lie on the unit circle except where
has repeated zeroes. The analogous statement is true for
; the zeroes of
will not lie on the critical line except where
has repeated zeroes.
Relating to this fact, it is a classical result of Speiser that the Riemann hypothesis is true if and only if all the zeroes of the derivative of the zeta function in the critical strip lie on or to the right of the critical line. The analogous result for polynomials is
Proposition 3 We have
(where all zeroes are counted with multiplicity.) In particular, the zeroes of
all lie on the unit circle if and only if the zeroes of
lie in the closed unit disk.
Proof: From the functional equation we have
Thus it will suffice to show that and
have the same number of zeroes outside the closed unit disk.
Set , then
is a rational function that does not have a zero or pole at infinity. For
not a zero of
, we have already seen that
and
are real, so on dividing we see that
is always real, that is to say
(This can also be seen by writing , where
runs over the zeroes of
, and using the fact that these zeroes are symmetric with respect to reflection across the unit circle.) When
is a zero of
,
has a simple pole at
with residue a positive multiple of
, and so
stays on the right half-plane if one traverses a semicircular arc around
outside the unit disk. From this and continuity we see that
stays on the right-half plane in a circle slightly larger than the unit circle, and hence by the argument principle it has the same number of zeroes and poles outside of this circle, giving the claim.
From the functional equation and the chain rule, is a zero of
if and only if
is a zero of
. We can thus write the above proposition in the equivalent form
One can use this identity to get a lower bound on the number of zeroes of by the method of mollifiers. Namely, for any other polynomial
, we clearly have
By Jensen’s formula, we have for any that
We therefore have
As the logarithm function is concave, we can apply Jensen’s inequality to conclude
where the expectation is over the parameter. It turns out that by choosing the mollifier
carefully in order to make
behave like the function
(while keeping the degree
small enough that one can compute the second moment here), and then optimising in
, one can use this inequality to get a positive fraction of zeroes of
on the unit circle on average. This is the polynomial analogue of a classical argument of Levinson, who used this to show that at least one third of the zeroes of the Riemann zeta function are on the critical line; all later improvements on this fraction have been based on some version of Levinson’s method, mainly focusing on more advanced choices for the mollifier
and of the differential operator
that implicitly appears in the above approach. (The most recent lower bound I know of is
, due to Pratt and Robles. In principle (as observed by Farmer) this bound can get arbitrarily close to
if one is allowed to use arbitrarily long mollifiers, but establishing this seems of comparable difficulty to unsolved problems such as the pair correlation conjecture; see this paper of Radziwill for more discussion.) A variant of these techniques can also establish “zero density estimates” of the following form: for any
, the number of zeroes of
that lie further than
from the unit circle is of order
on average for some absolute constant
. Thus, roughly speaking, most zeroes of
lie within
of the unit circle. (Analogues of these results for the Riemann zeta function were worked out by Selberg, by Jutila, and by Conrey, with increasingly strong values of
.)
The zeroes of tend to live somewhat closer to the origin than the zeroes of
. Suppose for instance that we write
where are the zeroes of
, then by evaluating at zero we see that
and the right-hand side is of unit magnitude by the functional equation. However, if we differentiate
where are the zeroes of
, then by evaluating at zero we now see that
The right-hand side would now be typically expected to be of size , and so on average we expect the
to have magnitude like
, that is to say pushed inwards from the unit circle by a distance roughly
. The analogous result for the Riemann zeta function is that the zeroes of
at height
lie at a distance roughly
to the right of the critical line on the average; see this paper of Levinson and Montgomery for a precise statement.
Important note: As this is not a course in probability, we will try to avoid developing the general theory of stochastic calculus (which includes such concepts as filtrations, martingales, and Ito calculus). This will unfortunately limit what we can actually prove rigorously, and so at some places the arguments will be somewhat informal in nature. A rigorous treatment of many of the topics here can be found for instance in Lawler’s Conformally Invariant Processes in the Plane, from which much of the material here is drawn.
In these notes, random variables will be denoted in boldface.
Definition 1 A real random variable
is said to be normally distributed with mean
and variance
if one has
for all test functions
. Similarly, a complex random variable
is said to be normally distributed with mean
and variance
if one has
for all test functions
, where
is the area element on
.
A real Brownian motion with base point
is a random, almost surely continuous function
(using the locally uniform topology on continuous functions) with the property that (almost surely)
, and for any sequence of times
, the increments
for
are independent real random variables that are normally distributed with mean zero and variance
. Similarly, a complex Brownian motion with base point
is a random, almost surely continuous function
with the property that
and for any sequence of times
, the increments
for
are independent complex random variables that are normally distributed with mean zero and variance
.
Remark 2 Thanks to the central limit theorem, the hypothesis that the increments
be normally distributed can be dropped from the definition of a Brownian motion, so long as one retains the independence and the normalisation of the mean and variance (technically one also needs some uniform integrability on the increments beyond the second moment, but we will not detail this here). A similar statement is also true for the complex Brownian motion (where now we need to normalise the variances and covariances of the real and imaginary parts of the increments).
Real and complex Brownian motions exist from any base point or
; see e.g. this previous blog post for a construction. We have the following simple invariances:
Exercise 3
- (i) (Translation invariance) If
is a real Brownian motion with base point
, and
, show that
is a real Brownian motion with base point
. Similarly, if
is a complex Brownian motion with base point
, and
, show that
is a complex Brownian motion with base point
.
- (ii) (Dilation invariance) If
is a real Brownian motion with base point
, and
is non-zero, show that
is also a real Brownian motion with base point
. Similarly, if
is a complex Brownian motion with base point
, and
is non-zero, show that
is also a complex Brownian motion with base point
.
- (iii) (Real and imaginary parts) If
is a complex Brownian motion with base point
, show that
and
are independent real Brownian motions with base point
. Conversely, if
are independent real Brownian motions of base point
, show that
is a complex Brownian motion with base point
.
The next lemma is a special case of the optional stopping theorem.
Lemma 4 (Optional stopping identities)
- (i) (Real case) Let
be a real Brownian motion with base point
. Let
be a bounded stopping time – a bounded random variable with the property that for any time
, the event that
is determined by the values of the trajectory
for times up to
(or more precisely, this event is measurable with respect to the
algebra generated by this proprtion of the trajectory). Then
and
and
- (ii) (Complex case) Let
be a real Brownian motion with base point
. Let
be a bounded stopping time – a bounded random variable with the property that for any time
, the event that
is determined by the values of the trajectory
for times up to
. Then
Proof: (Slightly informal) We just prove (i) and leave (ii) as an exercise. By translation invariance we can take . Let
be an upper bound for
. Since
is a real normally distributed variable with mean zero and variance
, we have
and
and
By the law of total expectation, we thus have
and
and
where the inner conditional expectations are with respect to the event that attains a particular point in
. However, from the independent increment nature of Brownian motion, once one conditions
to a fixed point
, the random variable
becomes a real normally distributed variable with mean
and variance
. Thus we have
and
and
which give the first two claims, and (after some algebra) the identity
which then also gives the third claim.
Exercise 5 Prove the second part of Lemma 4.
We now approach conformal maps from yet another perspective. Given an open subset of the complex numbers
, define a univalent function on
to be a holomorphic function
that is also injective. We will primarily be studying this concept in the case when
is the unit disk
.
Clearly, a univalent function on the unit disk is a conformal map from
to the image
; in particular,
is simply connected, and not all of
(since otherwise the inverse map
would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset
of the complex numbers is the image of a univalent function on
. Furthermore, if
contains the origin, then the univalent function
with this image becomes unique once we normalise
and
. Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions
with
and
. We will focus particular attention on the univalent functions
with the normalisation
and
; such functions will be called schlicht functions.
One basic example of a univalent function on is the Cayley transform
, which is a Möbius transformation from
to the right half-plane
. (The slight variant
is also referred to as the Cayley transform, as is the closely related map
, which maps
to the upper half-plane.) One can square this map to obtain a further univalent function
, which now maps
to the complex numbers with the negative real axis
removed. One can normalise this function to be schlicht to obtain the Koebe function
which now maps to the complex numbers with the half-line
removed. A little more generally, for any
we have the rotated Koebe function
that is a schlicht function that maps to the complex numbers with the half-line
removed.
Every schlicht function has a convergent Taylor expansion
for some complex coefficients with
. For instance, the Koebe function has the expansion
and similarly the rotated Koebe function has the expansion
Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients should obey the bound
for all
. After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.
It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions that are odd, thus
for all
, and the Taylor expansion now reads
for some complex coefficients with
. One can transform a general schlicht function
to an odd schlicht function
by observing that the function
, after removing the singularity at zero, is a non-zero function that equals
at the origin, and thus (as
is simply connected) has a unique holomorphic square root
that also equals
at the origin. If one then sets
it is not difficult to verify that is an odd schlicht function which additionally obeys the equation
Conversely, given an odd schlicht function , the formula (4) uniquely determines a schlicht function
.
For instance, if is the Koebe function (1),
becomes
which maps to the complex numbers with two slits
removed, and if
is the rotated Koebe function (2),
becomes
De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have
Theorem 1 (de Branges’ theorem) Let
be a natural number.
- (i) (Robertson conjecture) If
is an odd schlicht function, then
- (ii) (Bieberbach conjecture) If
is a schlicht function, then
It is easy to see that the Robertson conjecture for a given value of implies the Bieberbach conjecture for the same value of
. Indeed, if
is schlicht, and
is the odd schlicht function given by (3), then from extracting the
coefficient of (4) we obtain a formula
for the coefficients of in terms of the coefficients of
. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of
from the Robertson conjecture for the same value of
. We remark that Littlewood and Paley had conjectured a stronger form
of Robertson’s conjecture, but this was disproved for
by Fekete and Szegö.
To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of , known as the Milin conjecture. Next, one continuously enlarges the image
of the schlicht function to cover all of
; done properly, this places the schlicht function
as the initial function
in a sequence
of univalent maps
known as a Loewner chain. The functions
obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term
(or
, in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of
,
, or
. After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of
) but for which several proofs exist in the literature.
The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.
We now leave the topic of Riemann surfaces, and turn now to the (loosely related) topic of conformal mapping (and quasiconformal mapping). Recall that a conformal map from an open subset
of the complex plane to another open set
is a map that is holomorphic and bijective, which (by Rouché’s theorem) also forces the derivative of
to be nowhere vanishing. We then say that the two open sets
are conformally equivalent. From the Cauchy-Riemann equations we see that conformal maps are orientation-preserving and angle-preserving; from the Newton approximation
we see that they almost preserve small circles, indeed for
small the circle
will approximately map to
.
In previous quarters, we proved a fundamental theorem about this concept, the Riemann mapping theorem:
Theorem 1 (Riemann mapping theorem) Let
be a simply connected open subset of
that is not all of
. Then
is conformally equivalent to the unit disk
.
This theorem was proven in these 246A lecture notes, using an argument of Koebe. At a very high level, one can sketch Koebe’s proof of the Riemann mapping theorem as follows: among all the injective holomorphic maps from
to
that map some fixed point
to
, pick one that maximises the magnitude
of the derivative (ignoring for this discussion the issue of proving that a maximiser exists). If
avoids some point in
, one can compose
with various holomorphic maps and use Schwarz’s lemma and the chain rule to increase
without destroying injectivity; see the previous lecture notes for details. The conformal map
is unique up to Möbius automorphisms of the disk; one can fix the map by picking two distinct points
in
, and requiring
to be zero and
to be positive real.
It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Furthermore, one can run a version of Koebe’s argument (using now a discrete version of Perron’s method) to prove the Riemann mapping theorem through circle packings. In principle, this leads to a mostly elementary approach to conformal geometry, based on extremely classical mathematics that goes all the way back to Apollonius. However, in order to prove the basic existence and uniqueness theorems of circle packing, as well as the convergence to conformal maps in the continuous limit, it seems to be necessary (or at least highly convenient) to use much more modern machinery, including the theory of quasiconformal mapping, and also the Riemann mapping theorem itself (so in particular we are not structuring these notes to provide a completely independent proof of that theorem, though this may well be possible).
To make the above discussion more precise we need some notation.
Definition 2 (Circle packing) A (finite) circle packing is a finite collection
of circles
in the complex numbers indexed by some finite set
, whose interiors are all disjoint (but which are allowed to be tangent to each other), and whose union is connected. The nerve of a circle packing is the finite graph whose vertices
are the centres of the circle packing, with two such centres connected by an edge if the circles are tangent. (In these notes all graphs are undirected, finite and simple, unless otherwise specified.)
It is clear that the nerve of a circle packing is connected and planar, since one can draw the nerve by placing each vertex (tautologically) in its location in the complex plane, and drawing each edge by the line segment between the centres of the circles it connects (this line segment will pass through the point of tangency of the two circles). Later in these notes we will also have to consider some infinite circle packings, most notably the infinite regular hexagonal circle packing.
The first basic theorem in the subject is the following converse statement:
Theorem 3 (Circle packing theorem) Every connected planar graph is the nerve of a circle packing.
Of course, there can be multiple circle packings associated to a given connected planar graph; indeed, since reflections across a line and Möbius transformations map circles to circles (or lines), they will map circle packings to circle packings (unless one or more of the circles is sent to a line). It turns out that once one adds enough edges to the planar graph, the circle packing is otherwise rigid:
Theorem 4 (Koebe-Andreev-Thurston theorem) If a connected planar graph is maximal (i.e., no further edge can be added to it without destroying planarity), then the circle packing given by the above theorem is unique up to reflections and Möbius transformations.
Exercise 5 Let
be a connected planar graph with
vertices. Show that the following are equivalent:
- (i)
is a maximal planar graph.
- (ii)
has
edges.
- (iii) Every drawing
of
divides the plane into faces that have three edges each. (This includes one unbounded face.)
- (iv) At least one drawing
of
divides the plane into faces that have three edges each.
(Hint: use Euler’s formula
, where
is the number of faces including the unbounded face.)
Thurston conjectured that circle packings can be used to approximate the conformal map arising in the Riemann mapping theorem. Here is an informal statement:
Conjecture 6 (Informal Thurston conjecture) Let
be a simply connected domain, with two distinct points
. Let
be the conformal map from
to
that maps
to the origin and
to a positive real. For any small
, let
be the portion of the regular hexagonal circle packing by circles of radius
that are contained in
, and let
be an circle packing of
with all “boundary circles” tangent to
, giving rise to an “approximate map”
defined on the subset
of
consisting of the circles of
, their interiors, and the interstitial regions between triples of mutually tangent circles. Normalise this map so that
is zero and
is a positive real. Then
converges to
as
.
A rigorous version of this conjecture was proven by Rodin and Sullivan. Besides some elementary geometric lemmas (regarding the relative sizes of various configurations of tangent circles), the main ingredients are a rigidity result for the regular hexagonal circle packing, and the theory of quasiconformal maps. Quasiconformal maps are what seem on the surface to be a very broad generalisation of the notion of a conformal map. Informally, conformal maps take infinitesimal circles to infinitesimal circles, whereas quasiconformal maps take infinitesimal circles to infinitesimal ellipses of bounded eccentricity. In terms of Wirtinger derivatives, conformal maps obey the Cauchy-Riemann equation , while (sufficiently smooth) quasiconformal maps only obey an inequality
. As such, quasiconformal maps are considerably more plentiful than conformal maps, and in particular it is possible to create piecewise smooth quasiconformal maps by gluing together various simple maps such as affine maps or Möbius transformations; such piecewise maps will naturally arise when trying to rigorously build the map
alluded to in the above conjecture. On the other hand, it turns out that quasiconformal maps still have many vestiges of the rigidity properties enjoyed by conformal maps; for instance, there are quasiconformal analogues of fundamental theorems in conformal mapping such as the Schwarz reflection principle, Liouville’s theorem, or Hurwitz’s theorem. Among other things, these quasiconformal rigidity theorems allow one to create conformal maps from the limit of quasiconformal maps in many circumstances, and this will be how the Thurston conjecture will be proven. A key technical tool in establishing these sorts of rigidity theorems will be the theory of an important quasiconformal (quasi-)invariant, the conformal modulus (or, equivalently, the extremal length, which is the reciprocal of the modulus).
The fundamental object of study in real differential geometry are the real manifolds: Hausdorff topological spaces that locally look like open subsets of a Euclidean space
, and which can be equipped with an atlas
of coordinate charts
from open subsets
covering
to open subsets
in
, which are homeomorphisms; in particular, the transition maps
defined by
are all continuous. (It is also common to impose the requirement that the manifold
be second countable, though this will not be important for the current discussion.) A smooth real manifold is a real manifold in which the transition maps are all smooth.
In a similar fashion, the fundamental object of study in complex differential geometry are the complex manifolds, in which the model space is rather than
, and the transition maps
are required to be holomorphic (and not merely smooth or continuous). In the real case, the one-dimensional manifolds (curves) are quite simple to understand, particularly if one requires the manifold to be connected; for instance, all compact connected one-dimensional real manifolds are homeomorphic to the unit circle (why?). However, in the complex case, the connected one-dimensional manifolds – the ones that look locally like subsets of
– are much richer, and are known as Riemann surfaces. For sake of completeness we give the (somewhat lengthy) formal definition:
Definition 1 (Riemann surface) If
is a Hausdorff connected topological space, a (one-dimensional complex) atlas is a collection
of homeomorphisms from open subsets
of
that cover
to open subsets
of the complex numbers
, such that the transition maps
defined by
are all holomorphic. Here
is an arbitrary index set. Two atlases
,
on
are said to be equivalent if their union is also an atlas, thus the transition maps
and their inverses are all holomorphic. A Riemann surface is a Hausdorff connected topological space
equipped with an equivalence class of one-dimensional complex atlases.
A map
from one Riemann surface
to another
is holomorphic if the maps
are holomorphic for any charts
,
of an atlas of
and
respectively; it is not hard to see that this definition does not depend on the choice of atlas. It is also clear that the composition of two holomorphic maps is holomorphic (and in fact the class of Riemann surfaces with their holomorphic maps forms a category).
Here are some basic examples of Riemann surfaces.
Example 2 (Quotients of
) The complex numbers
clearly form a Riemann surface (using the identity map
as the single chart for an atlas). Of course, maps
that are holomorphic in the usual sense will also be holomorphic in the sense of the above definition, and vice versa, so the notion of holomorphicity for Riemann surfaces is compatible with that of holomorphicity for complex maps. More generally, given any discrete additive subgroup
of
, the quotient
is a Riemann surface. There are an infinite number of possible atlases to use here; one such is to pick a sufficiently small neighbourhood
of the origin in
and take the atlas
where
and
for all
. In particular, given any non-real complex number
, the complex torus
formed by quotienting
by the lattice
is a Riemann surface.
Example 3 Any open connected subset
of
is a Riemann surface. By the Riemann mapping theorem, all simply connected open
, other than
itself, are isomorphic (as Riemann surfaces) to the unit disk (or, equivalently, to the upper half-plane).
Example 4 (Riemann sphere) The Riemann sphere
, as a topological manifold, is the one-point compactification of
. Topologically, this is a sphere and is in particular connected. One can cover the Riemann sphere by the two open sets
and
, and give these two open sets the charts
and
defined by
for
,
for
, and
. This is a complex atlas since the
is holomorphic on
.
An alternate way of viewing the Riemann sphere is as the projective line
. Topologically, this is the punctured complex plane
quotiented out by non-zero complex dilations, thus elements of this space are equivalence classes
with the usual quotient topology. One can cover this space by two open sets
and
and give these two open sets the charts
and
defined by
for
,
. This is a complex atlas, basically because
for
and
is holomorphic on
.
Exercise 5 Verify that the Riemann sphere is isomorphic (as a Riemann surface) to the projective line.
Example 6 (Smooth algebraic plane curves) Let
be a complex polynomial in three variables which is homogeneous of some degree
, thus
Define the complex projective plane
to be the punctured space
quotiented out by non-zero complex dilations, with the usual quotient topology. (There is another important topology to place here of fundamental importance in algebraic geometry, namely the Zariski topology, but we will ignore this topology here.) This is a compact space, whose elements are equivalence classes
. Inside this plane we can define the (projective, degree
) algebraic curve
this is well defined thanks to (1). It is easy to verify that
is a closed subset of
and hence compact; it is non-empty thanks to the fundamental theorem of algebra.
Suppose that
is irreducible, which means that it is not the product of polynomials of smaller degree. As we shall show in the appendix, this makes the algebraic curve connected. (Actually, algebraic curves remain connected even in the reducible case, thanks to Bezout’s theorem, but we will not prove that theorem here.) We will in fact make the stronger nonsingularity hypothesis: there is no triple
such that the four numbers
simultaneously vanish for
. (This looks like four constraints, but is in fact essentially just three, due to the Euler identity
that arises from differentiating (1) in
. The fact that nonsingularity implies irreducibility is another consequence of Bezout’s theorem, which is not proven here.) For instance, the polynomial
is irreducible but singular (there is a “cusp” singularity at
). With this hypothesis, we call the curve
smooth.
Now suppose
is a point in
; without loss of generality we may take
non-zero, and then we can normalise
. Now one can think of
as an inhomogeneous polynomial in just two variables
, and by nondegeneracy we see that the gradient
is non-zero whenever
. By the (complexified) implicit function theorem, this ensures that the affine algebraic curve
is a Riemann surface in a neighbourhood of
; we leave this as an exercise. This can be used to give a coordinate chart for
in a neighbourhood of
when
. Similarly when
is non-zero. This can be shown to give an atlas on
, which (assuming the connectedness claim that we will prove later) gives
the structure of a Riemann surface.
Exercise 7 State and prove a complex version of the implicit function theorem that justifies the above claim that the charts in the above example form an atlas, and an algebraic curve associated to a non-singular polynomial is a Riemann surface.
- (i) Show that all (irreducible plane projective) algebraic curves of degree
are isomorphic to the Riemann sphere. (Hint: reduce to an explicit linear polynomial such as
.)
- (ii) Show that all (irreducible plane projective) algebraic curves of degree
are isomorphic to the Riemann sphere. (Hint: to reduce computation, first use some linear algebra to reduce the homogeneous quadratic polynomial to a standard form, such as
or
.)
Exercise 9 If
are complex numbers, show that the projective cubic curve
is nonsingular if and only if the discriminant
is non-zero. (When this occurs, the curve is called an elliptic curve (in Weierstrass form), which is a fundamentally important example of a Riemann surface in many areas of mathematics, and number theory in particular. One can also define the discriminant for polynomials of higher degree, but we will not do so here.)
A recurring theme in mathematics is that an object is often best studied by understanding spaces of “good” functions on
. In complex analysis, there are two basic types of good functions:
Definition 10 Let
be a Riemann surface. A holomorphic function on
is a holomorphic map from
to
; the space of all such functions will be denoted
. A meromorphic function on
is a holomorphic map from
to the Riemann sphere
, that is not identically equal to
; the space of all such functions will be denoted
.
One can also define holomorphicity and meromorphicity in terms of charts: a function is holomorphic if and only if, for any chart
, the map
is holomorphic in the usual complex analysis sense; similarly, a function
is meromorphic if and only if the preimage
is discrete (otherwise, by analytic continuation and the connectedness of
,
will be identically equal to
) and for any chart
, the map
becomes a meromorphic function in the usual complex analysis sense, after removing the discrete set of complex numbers where this map is infinite. One consequence of this alternate definition is that the space
of holomorphic functions is a commutative complex algebra (a complex vector space closed under pointwise multiplication), while the space
of meromorphic functions is a complex field (a commutative complex algebra where every non-zero element has an inverse). Another consequence is that one can define the notion of a zero of given order
, or a pole of order
, for a holomorphic or meromorphic function, by composing with a chart map and using the usual complex analysis notions there, noting (from the holomorphicity of transition maps and their inverses) that this does not depend on the choice of chart. (However, one cannot similarly define the residue of a meromorphic function on
this way, as the residue turns out to be chart-dependent thanks to the chain rule. Residues should instead be applied to meromorphic
-forms, a concept we will introduce later.) A third consequence is analytic continuation: if two holomorphic or meromorphic functions on
agree on a non-empty open set, then they agree everywhere.
On the complex numbers , there are of course many holomorphic functions and meromorphic functions; for instance any power series with an infinite radius of convergence will give a holomorphic function, and the quotient of any two such functions (with non-zero denominator) will give a meromorphic function. Furthermore, we have extremely wide latitude in how to specify the zeroes of the holomorphic function, or the zeroes and poles of the meromorphic function, thanks to tools such as the Weierstrass factorisation theorem or the Mittag-Leffler theorem (covered in previous quarters).
It turns out, however, that the situation changes dramatically when the Riemann surface is compact, with the holomorphic and meromorphic functions becoming much more rigid. First of all, compactness eliminates all holomorphic functions except for the constants:
Lemma 11 Let
be a holomorphic function on a compact Riemann surface
. Then
is constant.
This result should be seen as a close sibling of Liouville’s theorem that all bounded entire functions are constant. (Indeed, in the case of a complex torus, this lemma is a corollary of Liouville’s theorem.)
Proof: As is continuous and
is compact,
must attain a maximum at some point
. Working in a chart around
and applying the maximum principle, we conclude that
is constant in a neighbourhood of
, and hence is constant everywhere by analytic continuation.
This dramatically cuts down the number of possible meromorphic functions – indeed, for an abstract Riemann surface, it is not immediately obvious that there are any non-constant meromorphic functions at all! As the poles are isolated and the surface is compact, a meromorphic function can only have finitely many poles, and if one prescribes the location of the poles and the maximum order at each pole, then we shall see that the space of meromorphic functions is now finite dimensional. The precise dimensions of these spaces are in fact rather interesting, and obey a basic duality law known as the Riemann-Roch theorem. We will give a mostly self-contained proof of the Riemann-Roch theorem in these notes, omitting only some facts about genus and Euler characteristic, as well as construction of certain meromorphic -forms (also known as Abelian differentials).
A more detailed study of Riemann surface (and more generally, complex manifolds) can be found for instance in Griffiths and Harris’s “Principles of Algebraic Geometry“.
Next quarter (starting Monday, April 2) I will be teaching Math 246C (complex analysis) here at UCLA. This is the third in a three-series graduate course on complex analysis; a few years ago I taught the first course in this series (246A), so this course can be thought of in some sense as a sequel to that one (and would certainly assume knowledge of the material in that course as a prerequisite), although it also assumes knowledge of material from the second course 246B (which covers such topics as Weierstrass factorization and the theory of harmonic functions).
246C is primarily a topics course, and tends to be a somewhat miscellaneous collection of complex analysis subjects that were not covered in the previous two installments of the series. The initial topics I have in mind to cover are
- Elliptic functions;
- The Riemann-Roch theorem;
- Circle packings;
- The Bieberbach conjecture (proven by de Branges); and
- the Schramm-Loewner equation (SLE).
- This list is however subject to change (it is the first time I will have taught on any of these topics, and I am not yet certain on the most logical way to arrange them; also I am not completely certain that I will be able to cover all the above topics in ten weeks). I welcome reference recommendations and other suggestions from readers who have taught on one or more of these topics.
As usual, I will be posting lecture notes on this blog as the course progresses.
[Update: Mar 13: removed elliptic functions, as I have just learned that this was already covered in the prior 246B course.]
Let be a monic polynomial of degree
with complex coefficients. Then by the fundamental theorem of algebra, we can factor
as
for some complex zeroes (possibly with repetition).
Now suppose we evolve with respect to time by heat flow, creating a function
of two variables with given initial data
for which
On the space of polynomials of degree at most , the operator
is nilpotent, and one can solve this equation explicitly both forwards and backwards in time by the Taylor series
For instance, if one starts with a quadratic , then the polynomial evolves by the formula
As the polynomial evolves in time, the zeroes
evolve also. Assuming for sake of discussion that the zeroes are simple, the inverse function theorem tells us that the zeroes will (locally, at least) evolve smoothly in time. What are the dynamics of this evolution?
For instance, in the quadratic case, the quadratic formula tells us that the zeroes are
and
after arbitrarily choosing a branch of the square root. If are real and the discriminant
is initially positive, we see that we start with two real zeroes centred around
, which then approach each other until time
, at which point the roots collide and then move off from each other in an imaginary direction.
In the general case, we can obtain the equations of motion by implicitly differentiating the defining equation
in time using (2) to obtain
To simplify notation we drop the explicit dependence on time, thus
From (1) and the product rule, we see that
and
(where all indices are understood to range over ) leading to the equations of motion
at least when one avoids those times in which there is a repeated zero. In the case when the zeroes are real, each term
represents a (first-order) attraction in the dynamics between
and
, but the dynamics are more complicated for complex zeroes (e.g. purely imaginary zeroes will experience repulsion rather than attraction, as one already sees in the quadratic example). Curiously, this system resembles that of Dyson brownian motion (except with the brownian motion part removed, and time reversed). I learned of the connection between the ODE (3) and the heat equation from this paper of Csordas, Smith, and Varga, but perhaps it has been mentioned in earlier literature as well.
One interesting consequence of these equations is that if the zeroes are real at some time, then they will stay real as long as the zeroes do not collide. Let us now restrict attention to the case of real simple zeroes, in which case we will rename the zeroes as instead of
, and order them as
. The evolution
can now be thought of as reverse gradient flow for the “entropy”
(which is also essentially the logarithm of the discriminant of the polynomial) since we have
In particular, we have the monotonicity formula
where is the “energy”
where in the last line we use the antisymmetrisation identity
Among other things, this shows that as one goes backwards in time, the entropy decreases, and so no collisions can occur to the past, only in the future, which is of course consistent with the attractive nature of the dynamics. As is a convex function of the positions
, one expects
to also evolve in a convex manner in time, that is to say the energy
should be increasing. This is indeed the case:
Exercise 1 Show that
Symmetric polynomials of the zeroes are polynomial functions of the coefficients and should thus evolve in a polynomial fashion. One can compute this explicitly in simple cases. For instance, the center of mass is an invariant:
The variance decreases linearly:
Exercise 2 Establish the virial identity
As the variance (which is proportional to ) cannot become negative, this identity shows that “finite time blowup” must occur – that the zeroes must collide at or before the time
.
Exercise 3 Show that the Stieltjes transform
solves the viscous Burgers equation
either by using the original heat equation (2) and the identity
, or else by using the equations of motion (3). This relation between the Burgers equation and the heat equation is known as the Cole-Hopf transformation.
The paper of Csordas, Smith, and Varga mentioned previously gives some other bounds on the lifespan of the dynamics; roughly speaking, they show that if there is one pair of zeroes that are much closer to each other than to the other zeroes then they must collide in a short amount of time (unless there is a collision occuring even earlier at some other location). Their argument extends also to situations where there are an infinite number of zeroes, which they apply to get new results on Newman’s conjecture in analytic number theory. I would be curious to know of further places in the literature where this dynamics has been studied.
By an odd coincidence, I stumbled upon a second question in as many weeks about power series, and once again the only way I know how to prove the result is by complex methods; once again, I am leaving it here as a challenge to any interested readers, and I would be particularly interested in knowing of a proof that was not based on complex analysis (or thinly disguised versions thereof), or for a reference to previous literature where something like this identity has occured. (I suspect for instance that something like this may have shown up before in free probability, based on the answer to part (ii) of the problem.)
Here is a purely algebraic form of the problem:
Problem 1 Let
be a formal function of one variable
. Suppose that
is the formal function defined by
where we use
to denote the
-fold derivative of
with respect to the variable
.
- (i) Show that
can be formally recovered from
by the formula
- (ii) There is a remarkable further formal identity relating
with
that does not explicitly involve any infinite summation. What is this identity?
To rigorously formulate part (i) of this problem, one could work in the commutative differential ring of formal infinite series generated by polynomial combinations of and its derivatives (with no constant term). Part (ii) is a bit trickier to formulate in this abstract ring; the identity in question is easier to state if
are formal power series, or (even better) convergent power series, as it involves operations such as composition or inversion that can be more easily defined in those latter settings.
To illustrate Problem 1(i), let us compute up to third order in , using
to denote any quantity involving four or more factors of
and its derivatives, and similarly for other exponents than
. Then we have
and hence
multiplying, we have
and
and hence after a lot of canceling
Thus Problem 1(i) holds up to errors of at least. In principle one can continue verifying Problem 1(i) to increasingly high order in
, but the computations rapidly become quite lengthy, and I do not know of a direct way to ensure that one always obtains the required cancellation at the end of the computation.
Problem 1(i) can also be posed in formal power series: if
is a formal power series with no constant term with complex coefficients with
, then one can verify that the series
makes sense as a formal power series with no constant term, thus
For instance it is not difficult to show that . If one further has
, then it turns out that
as formal power series. Currently the only way I know how to show this is by first proving the claim for power series with a positive radius of convergence using the Cauchy integral formula, but even this is a bit tricky unless one has managed to guess the identity in (ii) first. (In fact, the way I discovered this problem was by first trying to solve (a variant of) the identity in (ii) by Taylor expansion in the course of attacking another problem, and obtaining the transform in Problem 1 as a consequence.)
The transform that takes to
resembles both the exponential function
and Taylor’s formula
but does not seem to be directly connected to either (this is more apparent once one knows the identity in (ii)).
In the previous set of notes we introduced the notion of a complex diffeomorphism between two open subsets
of the complex plane
(or more generally, two Riemann surfaces): an invertible holomorphic map whose inverse was also holomorphic. (Actually, the last part is automatic, thanks to Exercise 40 of Notes 4.) Such maps are also known as biholomorphic maps or conformal maps (although in some literature the notion of “conformal map” is expanded to permit maps such as the complex conjugation map
that are angle-preserving but not orientation-preserving, as well as maps such as the exponential map
from
to
that are only locally injective rather than globally injective). Such complex diffeomorphisms can be used in complex analysis (or in the analysis of harmonic functions) to change the underlying domain
to a domain that may be more convenient for calculations, thanks to the following basic lemma:
Lemma 1 (Holomorphicity and harmonicity are conformal invariants) Let
be a complex diffeomorphism between two Riemann surfaces
.
- (i) If
is a function to another Riemann surface
, then
is holomorphic if and only if
is holomorphic.
- (ii) If
are open subsets of
and
is a function, then
is harmonic if and only if
is harmonic.
Proof: Part (i) is immediate since the composition of two holomorphic functions is holomorphic. For part (ii), observe that if is harmonic then on any ball
in
,
is the real part of some holomorphic function
thanks to Exercise 62 of Notes 3. By part (i),
is also holomorphic. Taking real parts we see that
is harmonic on each ball
in
, and hence harmonic on all of
, giving one direction of (ii); the other direction is proven similarly.
Exercise 2 Establish Lemma 1(ii) by direct calculation, avoiding the use of holomorphic functions. (Hint: the calculations are cleanest if one uses Wirtinger derivatives, as per Exercise 27 of Notes 1.)
Exercise 3 Let
be a complex diffeomorphism between two open subsets
of
, let
be a point in
, let
be a natural number, and let
be holomorphic. Show that
has a zero (resp. a pole) of order
at
if and only if
has a zero (resp. a pole) of order
at
.
From Lemma 1(ii) we can now define the notion of a harmonic function on a Riemann surface
; such a function
is harmonic if, for every coordinate chart
in some atlas, the map
is harmonic. Lemma 1(ii) ensures that this definition of harmonicity does not depend on the choice of atlas. Similarly, using Exercise 3 one can define what it means for a holomorphic map
on a Riemann surface
to have a pole or zero of a given order at a point
, with the definition being independent of the choice of atlas.
In view of Lemma 1, it is thus natural to ask which Riemann surfaces are complex diffeomorphic to each other, and more generally to understand the space of holomorphic maps from one given Riemann surface to another. We will initially focus attention on three important model Riemann surfaces:
- (i) (Elliptic model) The Riemann sphere
;
- (ii) (Parabolic model) The complex plane
; and
- (iii) (Hyperbolic model) The unit disk
.
The designation of these model Riemann surfaces as elliptic, parabolic, and hyperbolic comes from Riemannian geometry, where it is natural to endow each of these surfaces with a constant curvature Riemannian metric which is positive, zero, or negative in the elliptic, parabolic, and hyperbolic cases respectively. However, we will not discuss Riemannian geometry further here.
All three model Riemann surfaces are simply connected, but none of them are complex diffeomorphic to any other; indeed, there are no non-constant holomorphic maps from the Riemann sphere to the plane or the disk, nor are there any non-constant holomorphic maps from the plane to the disk (although there are plenty of holomorphic maps going in the opposite directions). The complex automorphisms (that is, the complex diffeomorphisms from a surface to itself) of each of the three surfaces can be classified explicitly. The automorphisms of the Riemann sphere turn out to be the Möbius transformations with
, also known as fractional linear transformations. The automorphisms of the complex plane are the linear transformations
with
, and the automorphisms of the disk are the fractional linear transformations of the form
for
and
. Holomorphic maps
from the disk
to itself that fix the origin obey a basic but incredibly important estimate known as the Schwarz lemma: they are “dominated” by the identity function
in the sense that
for all
. Among other things, this lemma gives guidance to determine when a given Riemann surface is complex diffeomorphic to a disk; we shall discuss this point further below.
It is a beautiful and fundamental fact in complex analysis that these three model Riemann surfaces are in fact an exhaustive list of the simply connected Riemann surfaces, up to complex diffeomorphism. More precisely, we have the Riemann mapping theorem and the uniformisation theorem:
Theorem 4 (Riemann mapping theorem) Let
be a simply connected open subset of
that is not all of
. Then
is complex diffeomorphic to
.
Theorem 5 (Uniformisation theorem) Let
be a simply connected Riemann surface. Then
is complex diffeomorphic to
,
, or
.
As we shall see, every connected Riemann surface can be viewed as the quotient of its simply connected universal cover by a discrete group of automorphisms known as deck transformations. This in principle gives a complete classification of Riemann surfaces up to complex diffeomorphism, although the situation is still somewhat complicated in the hyperbolic case because of the wide variety of discrete groups of automorphisms available in that case.
We will prove the Riemann mapping theorem in these notes, using the elegant argument of Koebe that is based on the Schwarz lemma and Montel’s theorem (Exercise 57 of Notes 4). The uniformisation theorem is however more difficult to establish; we discuss some components of a proof (based on the Perron method of subharmonic functions) here, but stop short of providing a complete proof.
The above theorems show that it is in principle possible to conformally map various domains into model domains such as the unit disk, but the proofs of these theorems do not readily produce explicit conformal maps for this purpose. For some domains we can just write down a suitable such map. For instance:
Exercise 6 (Cayley transform) Let
be the upper half-plane. Show that the Cayley transform
, defined by
is a complex diffeomorphism from the upper half-plane
to the disk
, with inverse map
given by
Exercise 7 Show that for any real numbers
, the strip
is complex diffeomorphic to the disk
. (Hint: use the complex exponential and a linear transformation to map the strip onto the half-plane
.)
Exercise 8 Show that for any real numbers $latex {a<b0, a < \theta < b \}}&fg=000000$ is complex diffeomorphic to the disk
. (Hint: use a branch of either the complex logarithm, or of a complex power
.)
We will discuss some other explicit conformal maps in this set of notes, such as the Schwarz-Christoffel maps that transform the upper half-plane to polygonal regions. Further examples of conformal mapping can be found in the text of Stein-Shakarchi.
My colleague Tom Liggett recently posed to me the following problem about power series in one real variable . Observe that the power series
has very rapidly decaying coefficients (of order ), leading to an infinite radius of convergence; also, as the series converges to
, the series decays very rapidly as
approaches
. The problem is whether this is essentially the only example of this type. More precisely:
Problem 1 Let
be a bounded sequence of real numbers, and suppose that the power series
(which has an infinite radius of convergence) decays like
as
, in the sense that the function
remains bounded as
. Must the sequence
be of the form
for some constant
?
As it turns out, the problem has a very nice solution using complex analysis methods, which by coincidence I happen to be teaching right now. I am therefore posing as a challenge to my complex analysis students and to other readers of this blog to answer the above problem by complex methods; feel free to post solutions in the comments below (and in particular, if you don’t want to be spoiled, you should probably refrain from reading the comments). In fact, the only way I know how to solve this problem currently is by complex methods; I would be interested in seeing a purely real-variable solution that is not simply a thinly disguised version of a complex-variable argument.
(To be fair to my students, the complex variable argument does require one additional tool that is not directly covered in my notes. That tool can be found here.)
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