A geometric version of the Schwarz lemma
Gunnar Þór Magnússon
A talk given at KAUS 2010
Abstract
Lars Alhfors proved a differential geometric version of the classical
Schwarz lemma in 1938. His version of the lemma gives an interesting
connection between the existence of non-constant entire functions with
values in a given domain and metrics with negative curvature on said
domain. We recall the classical Schwarz lemma and review the notions
necessary to understand Ahlfors’ lemma, before proving both the new
form of the lemma and giving some applications.
1.
The Schwarz lemma
Let’s start by recalling the classical form of the Schwarz lemma.
Lemma. (Schwarz) Let f : D → D be holomorphic and suppose that
f (0) = 0. Then |f (z)| ≤ |z| for all z and |f 0 (0)| ≤ 1, with equality in either
case if and only if f is a rotation.
Proof: The function g(z) := f (z)/z is holomorphic on D. Set M (r) =
supD(0,r) |g(z)| for 0 ≤ r < 1, note that M (r) is bounded above by 1/r by
the maximum principle, and let r → 1. Equality in either case forces g to be
constant of modulus 1.
Definition. A Möbius transformation is a map ϕa : D → C of the form
ϕa (z) =
z−a
1 − az
where a ∈ D.
Proposition. Let ϕa be a Möbius transformation. Then ϕa is holomorphic, ϕa (a) = 0, and ϕa sends the disk to itself. Furthermore ϕa is bijective
and its inverse is given by ϕ−a .
Proof: The first two claims are trivial. Note that ϕa extends continuously to D. If |z| = 1, then a calculation shows that |ϕa (z)| = 1, so the
maximum principle implies that the image of the disk lies in the disk. For
the last claim, check that ϕ−a ◦ ϕa fixes both 0 and a and apply the Schwarz
lemma.
Recall that if f : U → V is holomorphic and bijective, then its inverse
is holomorphic as well. If f : U → U is holomorphic and bijective, then
we call f an automorphism of U . The set of all automorphisms of U is a
group under the composition of functions.
f −1
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2
METRICS AND CURVATURE
Proposition. Let f be an automorphism of the disk. Then f is the
composition of a Möbius transformation and a rotation.
Proof: Set a = f −1 (0) and g := f ◦ ϕ−a . Then g and g −1 are automorphisms of the disk that fix 0. We find that (g −1 )0 (0) = 1/g 0 (0), so g is a
rotation by the Schwarz lemma.
The condition that f (0) = 0 in the Schwarz lemma is restrictive, but we
can remove it with the aid of a Möbius transformation:
Lemma. (Scwartz-Pick) Let f : D → D be holomorphic. For any z 6= w
we have
f (w) − f (z) w − z ≤
1 − f (w)f (z) 1 − wz and
|f 0 (z)|
1
≤
2
1 − |f (z)|
1 − |z|2
with equality in either case if and only if f is an automorphism of the disk.
Proof: For the first inequality, compose f with suitably chosen Möbius
transformations and apply the classical Schwarz lemma. For the second, let
w → z. The statements about equality follow from the composition of f
with a Möbius transformation.
2.
Metrics and curvature
We will now briefly review some basic notions of Riemannian geometry.
Metrics. Let U ⊂ C be a domain, that is a connected open set. A
metric on U is a C 2 function σ : U → R+ . We allow σ either to have
isolated zeros or be identically zero, though all the metrics we shall see in
practice will be strictly positive. For a vector ξ ∈ TU,z = C we define
|ξ|σ,z := σ(z) |ξ|
where |ξ| denotes the euclidean length of ξ. The length of a tangent vector
thus depends on its position in U .
Pullbacks. Let U and V be domains in C and let f : U → V be a
holomorphic function. Let σ be a metric on V . The pullback of σ to U is
f ∗ σ(z) := |f 0 (z)| σ(f (z)).
(1)
If f 0 is not identically zero then f 0 has only isolated zeros, so f ∗ σ is a metric
on U .
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2
METRICS AND CURVATURE
¯ Let σ be a metric
Curvature. Recall that the Laplacian is ∆ = 4∂ ∂.
on a domain U . The curvature of σ is
κ(z) =
−∆ log σ(z)
σ(z)2
for those z such that σ(z) is nonzero, it is undefined otherwise. Curvature
is invariant under the change of holomorphic coordinates:
Proposition 2.1. Let f : U → V be biholomorphic and let σ be a
metric on V . The curvature of f ∗ σ satisfies
κf ∗ σ (z) = κσ (f (z))
(2)
Proof: Set g(z) := log(σ(z)). As f is holomorphic the chain rule for the
∂ and ∂¯ operators gives
¯ ◦ f )(z)
∂ ∂¯ log(σ ◦ f )(z) = ∂ ∂(g
¯ (z))
= ∂ ∂¯f¯ · ∂g(f
¯ (z)) + ∂f · ∂f · ∂ ∂g(f
¯ (z))
= ∂ ∂¯f¯ · ∂g(f
= |f 0 (z)|2 · ∂ ∂¯ log σ (f (z)),
because f¯ is harmonic. As f is biholomorphic its derivative is nowhere zero,
so
κf ∗ σ (z) =
−|f 0 (z)|2 ∆ log σ(f (z))
−∆ log(σ ◦ f )(z)
=
= κσ (f (z))
(σ ◦ f (z))2
|f 0 (z)|2 σ(f (z))2
like we wanted.
Remark — In the proof of this last proposition we only needed to use
that f was holomorphic and that its derivative was nowhere zero. The proposition thus holds for any such function. In particular, the curvature of a
metric is invariant under the pullback of a local biholomorphism, or an umramified covering map. We will use this fact later.
Our interest in this discussion will mostly be in one metric in particular:
Example. Let D := D(0, 1) = {z ∈ C | |z| < 1} be the unit disk in C.
The Poincaré metric is defined as
ρ(z) =
1
1 − |z|2
We calculate that ∆ log ρ = 4ρ2 , so the Poincaré metric has constant curvature −4. Note that the Schwarz-Pick lemma says the Poincaré metric is
invariant under the pullback of any automorphism of the disk.
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3
ALHFORS
Remark — We extend the definition of a metric to a Riemann surface by
demanding that any such metric σ satisfies (1) for any change of coordinates
f : U → V . The curvature of a metric is defined locally and depends a priori
on the coordinates chosen, but Proposition 2.1 tells us that it is well defined
on the whole surface.
3.
Alhfors
The language we have introduced allows us to formulate the Schwarz-Pick
lemma in very concise terms:
Proposition. Let f : D → D be a holomorphic function. Then f is
distance decreasing in the Poincaré metric, i.e.
f ∗ ρ(z) ≤ ρ(z),
for all z in the disk D.
There is nothing to prove here, we only need to unpack the definitions involved to see that this is just a restatement of the Schwarz-Pick lemma. This
restatement does however point the way to a new version of the Schwarz
lemma:
Theorem 3.1. (Ahlfors) Let ρ be the Poincaré metric on D, let S be
a Riemann surface with a metric σ, and let f : D → S be holomorphic. If
there is a constant B > 0 such that κσ ≤ −B < 0 at all points of S, then
2
f ∗σ ≤ √ ρ
B
Proof: Take 0 < r < 1 and let
2
ρr := √ (z/r)∗ ρ.
B
Then ρr is a metric on D(0, r) with constant curvature −B. Define
(
v : D(0, r) → R
∗
z 7−→ fρrσ(z)
(z)
Note that v is positive and continuous. Also, as D(0, r) ⊂⊂ D then f ∗ σ is
bounded from above on D(0, r), while ρ(z) → +∞ as |z| approaches r, so
v(z) → 0 as |z| tends to r. It follows that v attains a maximum M at a point
z0 in D(0, r).
If f ∗ σ(z0 ) = 0 then v ≡ 0 and the result is certainly true. Suppose thus
that f ∗ σ(z0 ) > 0, so κf ∗ σ is defined at z0 . Now, log v has a maximum at z0 ,
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4
HYPERBOLICITY
so
0 ≥ ∆ log v(z0 ) = ∆ log f ∗ σ(z0 ) − ∆ log ρr (z0 )
= f ∗ σ(z0 )2 κf ∗ σ (z0 ) − ρr (z0 )2 κρr (z0 )
≥ B(f ∗ σ(z0 )2 − ρr (z0 )2 ),
which gives ρr (z0 ) ≥ f ∗ σ(z0 ), or M ≤ 1 on D(0, r). The upper bound M ≤ 1
on v is independent of r, and we conclude that f ∗ σ ≤ ρ on D.
As it turns out, Ahlfors’ theorem gives a very interesting connection
between the existence of non-constant holomorphic functions and metrics
with negative curvature:
Theorem 3.2. Let S be a Riemann surface which admits a metric σ
with curvature κσ ≤ −B < 0. Then any holomorphic function f : C → S is
constant.
Proof: Fix z0 in C. For any r > 0 such that z0 ∈ D(0, r), Theorem 3.1
gives f ∗ σ(z0 ) ≤ ρr (z0 ), where ρr is as in the proof of Theorem 3.1. Letting
r tend to infinity we get
0 ≤ f ∗ σ(z0 ) ≤ 0,
so f ∗ σ(z0 ) = 0. But z0 was arbitrary, so f ∗ σ = 0.
Now take a coordinate neighborhood V of f (z0 ) in S, let ψ : V → W ⊂ C
be a coordinate function on V , and set U = f −1 (V ). If g := ψ ◦ f , then the
condition f ∗ σ = 0 gives that g 0 = 0 on U , i.e. that g is constant on U . We
conclude that f is constant on the open set U , and thus on the whole of C.
As a corollary we find the well known:
Theorem. (Liouville) Let f : C → C be bounded. Then f is constant.
Proof: If M > |f (z)| for all z in C, then f /M takes its values in the
disk which admits the Poincaré metric of constant curvature −4.
4.
Hyperbolicity
Taking our cue from the results in the last section we now define:
Definition. A Riemann surface is hyperbolic if it admits a metric σ
with curvature κσ ≤ −B < 0.
Recall that given a Riemann surface S, we can construct a simply connected Riemann surface S̃ and a holomorphic covering map π : S̃ → S. The
Riemann surface S̃ is the universal covering of S.
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4
HYPERBOLICITY
Suppose that S admits a hyperbolic metric σ. Then the pullback π ∗ σ
gives a metric on the universal covering S̃. As S̃ is locally biholomorphic to
S the curvature of this new metric satisfies
κπ∗ σ (z) = κσ (π(z)) ≤ −B < 0,
so S̃ is hyperbolic as well.
At this point, it is natural to ask oneself which simply connected Riemann
surfaces are hyperbolic. We can obtain the answer from the:
Uniformization theorem. Any simply connected Riemann surface is
isomorphic to either the unit disk D, the complex plane C, or the projective
line P1 .
Note that there exist non-constant entire maps on C, and non constant
holomorphic maps f : C → P1 (the imbedding C ,→ P1 , for example). Thus
by Theorem 3.2 the unit disk D is the only simply connected hyperbolic
Riemann surface.
Now, if S is a Riemann surface and S̄ is its universal covering, then
S ' S̃/G, where G is a properly discontinuous and fixed-point free subgroup
of the automorphism group of S̃. Suppose that S̃ is the unit disk D, i.e. that
the universal covering of S is hyperbolic. Does it follow that S itself is
hyperbolic?
The unit disk D admits the Poincaré metric ρ, which has constant curvature −4 and is invariant under any automorphism of the disk. Our Riemann
surface S is biholomorphic to S̃/G, where G is a subgroup of automorphisms
of D. It follows that ρ gives rise to a metric σ on S:
σ(π(z)) := ρ(z),
where the metric σ is well defined because π is surjective and ρ is invariant
under G. What is the curvature of this metric?
The map π is a local biholomorphism. Locally the metric σ agrees with
the pullback of ρ to S via π −1 , so its curvature satisfies
κσ (w) = κπ−1 ∗ ρ (w) = κρ (π −1 (w)) = −4,
and we conclude that S is hyperbolic. We have proved:
Theorem. A Riemann surface S is hyperbolic if and only if its universal covering is the unit disk D.
Remark — 1) Recall that any automorphism of the projective line P1 is
a Möbius transformation of the form f (z) = az+b
cz+d with ad − bc 6= 0. Such a
map necessarily has a fixed point (solve f (z) = z), thus Aut P1 contains no
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5
APPLICATIONS
subgroup of fixed point free morphisms. That is, the only Riemann surface
which admits P1 as its universal cover is P1 itself.
2) The group of automorphisms of the complex plane is the group of
affine transformations f (z) = az + b, where a 6= 0. Aside from the identity,
no such map has a fixed point. However, we can prove that any properly
discontinuous subgroup G of Aut C is either trivial, or cyclic with one or
two generators. It follows that any Riemann surface S with S̃ = C is either
C itself, the punctured plane C∗ , or a torus.
3) We conclude that any Riemann surface S which is not a torus, C∗ ,
the complex plane or the projective line is hyperbolic.
Remark — For any hyperbolic Riemann surface S, the proof of the above
theorem gives√a metric σ on S with constant curvature −4. Multiplying this
metric by 2/ B we obtain a metric on S with constant curvature −B, for
any B > 0. Thus a Riemann surface is hyperbolic if and only if it admits a
metric of constant negative curvature.
5.
Applications
Recall that we can assign a unique integer, which we call genus, to any
Riemann surface. It is a topological invariant. Intuitively the genus of a
Riemann surface is the number of holes in it. From the discussion in the
last section, we get in particular that any Riemann surface of genus g ≥ 2
is hyperbolic, and we obtain an entertaining proof of:
Theorem. (Picard) Any holomorphic f : C → C which omits two distinct values a and b is constant.
Proof: The function f takes its values in C\{a, b}, which has two holes.
The case of elliptic curves. Note that if E is an elliptic curve (a
complex torus of dimension 1), S is a Riemann surface and f : E → S is
holomorphic, then f extends to a holomorphic f¯ : C → S. We can thus apply
Theorem 3.2 to tori. Note that the genre remarks above apply in particular
to compact Riemann surfaces, so we immediately obtain:
Proposition. If E is an elliptic curve and S is a compact Riemann
surface of genre g ≥ 2, then any holomorphic map f : E → S is constant.
This seems amusing, as I at least know of no easy proof of the same fact which
uses more algebraic machinery or techniques from the theory of compact
Riemann surfaces.
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5
APPLICATIONS
References
1. Ahlfors, Lars V. An extension of Schwartz’s lemma. Transactions
of the American Mathematical Society, Vol. 43, No. 3 (May, 1938),
pp. 359-364.
2. Jost, Jürgen. Compact Riemann Surfaces. 3rd edition. Springer,
2000.
3. Krantz, Steven G. Geometric function theory. Birkhauser, 2006.
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