manuscripta
math.
72,
251
- 256
(1991)
manuscripta
mathematica
9 Springer-Verlag1991
T H E S C H W A R Z L E M M A F O R N O N P O S I T I V E L Y CURVED
R I E M A N N I A N SURFACES
Marc Troyanov
In this paper, we prove that if f is a conformal map between two Riemannian
surfaces, and if the curvature of the target is nonpositive and less than or equal
to the curvature of the source, then the map is contracting.
The well known Schwarz lemma of complex analysis has been generalized
for KKhler manifolds by Yau in [7]. In the special case of Riemann surfaces, his
result can be stated in the following way :
T h e o r e m [7]. Let (Sl,gl) and ($2,g2) be two smooth Riemannian surfaces
without boundary. Let Ki denote the curvature of (Si,g~) and assume the
following :
a)
( Sl , gl ) is complete ;
b)
K1 > - a l for some number al > 0 ;
c)
K2 <_ -a~ < 0 .
Then any conformM map f : $1 ~ $2 satisfies :
f*(g2) <_ ( ~ ) g l .
In particular, if am = O, then f is locally constant.
When (51,91) is the hyperbolic plane, this result has already been proved
by L.V. Ahlfors in 1938 (see [1]).
In Yau's theorem, we have to assume the target surface to have curvature
bounded above by a negative constant. Our aim in this paper is to prove a
Schwarz lemma for surfaces of non positive curvature.
251
MARC TROYANOV
T h e o r e m ( T h e S e h w a r z L e m m a ) . Let (Sz,gl) be a smooth complete connected Riemannian surface with curvature K1 bounded below and ($2,g2) be
any smooth Riemannian surface with curvature K2.
Let f : (SI, gl) -* ($2, g2) be a conformal mapping such that
K20f
_<min{0, K z } ,
Ks 0 f
< -a < 0
K2of~O
and
on the complement of some compact subset of $1 .
Then f is a contracting map.
The proof of this theorem will rely on the generalized m a x i m u m principle of
Yan. But let us show first how we may obtain simple proofs of some results in
the geometry of surfaces using the Schwarz lemma.
T h e o r e m ( L i o u v i l l e ' s T h e o r e m ) . Let (St,g1) be a smooth complete connected Riemannian surface with curvature K1 > O, and ($2, g2) be any smooth
Riemannian surface with curvature Ks.
Then any conformal map f : $1 -* S2 such that
K2of
<_0,
K2of~O
Ks o f
< -a < 0
and
on the complement of some compact subset of $1 .
is constant.
Proof. The map f : ($1, cgl) -* ($2,g2) has to be contracting for any
positive constant c. []
Let us now discuss some applications to Riemann surface theory. Recall
first that a Riemann surface is a one dimensional complex manifold and that a
Riemannian surface is 2-dimensional manifold with a Riemannian metric on it.
A metric g on a Pdemann surface S is said to be conformal if, on any domain
U C S of a complex coordinate z, the metric takes the form g = p(z)[dz] 2
(for some positive function p). Conversely, given a Riemannian metric g on an
oriented surface S, there exists a unique complex structure on S for which this
metric is conformal (Korn-Lichtenchtein theorem). Furthermore, a mapping
f : ($1, gt) --* (S~,g2) between oriented Riemannian surfaces is conformal if
and only if it is holomorphic for the associated complex structure.
T h e complex line is denoted by C, and the once punctured complex line by
C*.
C o r o l l a r y 1. (Compare to [5].) There is no conformal metric g on C or C*
(whether complete or not) such that Kg <_ 0 and K 9 <_ - a < 0 outside a
compact set.
Proof. Suppose that such a metric g exists on S = C or C*. Let go be the
canonical (conformal, complete and euclidean) metric on S. then the identity
map id : (S, go) -* (S, g) is constant, which is absurd. []
Observe t h a t this result is false on any other open Pdemann surface of finite
topological type (see [3], th. A.1).
252
MARC TROYANOV
C o r o l l a r y 2 ( P i c a r d ' s T h e o r e m ) . Let f be a non constant entire function
(holomorphic function on C). Then C \ f ( C ) contains at most one point.
Proof Let $2 be C minus two points. Then S~ carries a conformal metric
92 of curvature -1. Suppose that f is an entire function whose image lies in S~.
Then the function f is a conformal map from C --* $2, hence a constant. []
Our next application of the Schwarz lemma says that a complete conformal
metric on a Riemann surface is determined by its curvature (if the curvature
is negative).
T h e o r e m . (Uniqueness of the solution to the Berger-Nirenberg Problem.)
Let S be an open R/emann surface (whithout boundary and of/inite topological type) and K : S --* t t be a smooth non positive function such that
-b<_K <_-a<O
on the complement of some compact set. Assume that g and h are two complete
conformal metrics on S with curvature K. Then g = h.
Proof. Apply the Schwarz lemma to the identity map (S, h) --* (S, g) and
its inverse. []
The existence of metric with prescribed curvature on a given open Riemann
surface (of finite type) is dicussed in [2], [3].
For an application of the higher dimensional Schwarz lemma, see [4].
In the proof of the Schwarz lemma, we will need the following definition; let
/~ he some real number such that/~ > - 1 .
Delinition : The conformal metric g on the Riemann surface S has a conical
singularity of order/~ at p E S if there is a local complex coordinate z centered
at p such that we have the local expression
where v is a continuous function (this notion is independent of the choice of
the coordinate z).
We will always assume our metrics to be smooth except perhaps for isolated
conical singularities. In particular, the curvature Kg of such a metric g is a well
defined continuous function off the singular set. Inequalities such as Kg <_ 0
will always be interpreted on the complement of the singular set.
Example. If $1 and $2 are two Riemann surfaces, g is a smooth conformal
metric on $2 and f : 5'1 ---, $2 is a holomorphic map having a zero of order m
at p E $1. Then f*g obviously has a conical singularity of order m - 1 at p.
253
MARC TROYANOV
Proof
of the Sehwarz
Lemma.
If f is constant, then there is nothing to prove, we m a y therefore assume
t h a t 3* is a non constant holomorphic m a p with isolated singularities.
Let P l , P 2 , " " E S1 be the list of all singular points of f and let S~ :=
Sa \ { p l , p 2 , - . . }. T h e n we m a y define a function u on S~ by f*(g2) = e2"gl.
We want to prove t h a t u < 0.
In the neighbourhood of pi, we have in some local coordinate z (with
z(p,) = 0)
f ( z ) = z m' ,
where mi > 1 is some positive integer, therefore, u(z) = vi(z) + (mi - 1) log(z)
for some locally defined continuous function vi (in other words, pi is a conical
singularity of order mi - 1 for the metric f*(g2)). We can thus find small
neighbourhoods/4i of the Pi'S and a s m o o t h function ~ on S] such t h a t
u<fi<-I
on // for all i ,
u=
on
~
Sl \/4 ,
w h e r e / 4 : = Ui/4i.
O n the complement of the set U, the function u is s m o o t h and we have
Au=(K2of)e
x"-K1
onS\/4,
on O(S \ / 4 )
u < -1
,
where A is the Laplacian for the metric gl (with the sign convention t h a t a
subharmonic function to satisfies At0 _< 0).
Consider now the set 7) = {x E S : u(x) >_ 0}. It is clear t h a t 7) n / 4 = 0, in
particular, u = fi on 7). Observe also t h a t u is subharmonic on 7) since
A u = ( K = o f ) e 2u - K 1 = ( K 2 o f ) ( e 2u -
1) + ((K2 o f ) - K1)
_< (K2 o f ) ( e 2u - 1) < 0.
Case 1 : u achieves its m a x i m u m and 07 ) r 0.
T h e m a x i m u m principle tells us t h a t m a x u is attained on 0 P . But it follows
from the definitions of 7) t h a t u ~ 0 on 0 P . Hence m a x u _< 0 on S.
Case 2 : u achieves its m a x i m u m and ~1~ = 0.
T h e n either 7) = 0 or 7) = S. In the latter case, u is suhharmonic on S and
hence is a constant. It is easy to see t h a t the hypothesis on K1 and (K2 o f )
imply t h a t this constant is non positive.
Case 3 : u does not achieve its m a x i m u m . T h e n either 7) = 0, or we can
find r / > 0 and z E S such t h a t
u(=) > ,7.
254
MARC TROYANOV
We shall use the generalized m a x i m u m principle of Yau (see t h e o r e m 1, p.
200 in [7], or t h e o r e m 1, p. 206 in [Y1]). It says t h a t if v is a C 2 function
on a complete manifold (Sl,gl) with (Ricci) curvature b o u n d e d below, and if
sup(v) < r is not attained, then for any e, ~ > 0, the set
(*)
f2 = {z E S \ N : A v ( z ) > - e , IVv(=)l 2 < ,
and v(x) > sup(v) - 6}
is not e m p t y where N C S is any given c o m p a c t set.
We will apply this principle to the b o u n d e d function v : St --~ R defined by
1
v(z) = 1 + e-~(~)"
Choose ~ > 0 small enough so that u(z) > 7 / > 0 for v(z) > sup(v) - 6 (i.e.
/~ < sup(v) - (1 + e - ~ ) - l ) . Choose N C $1 to be the (compact) set of points
in S1 such that K~ o f > - a . Finally, choose e so that
4e <
-
asinh(~/)
1 + 2sinh(~) "
A n d let f2 be the set defined by (*) with these choices of v,e, 6 and N.
A c o m p u t a t i o n gives us
e-U
(1 + e-U) ~
Au = Av - 2sinh (u)lVvl 2 .
Since u > 0 on f2, we have (1 + e-U) 2 < 4
on A v and IVy[ 2, we see t h a t
on f2, using also the estimates
e - U A u >__- 4 e ( 1 + 2sinh(u))
on D.
From A u = (K2 o .[)e 2~ - K1, we get
(K2 o f ) e u - K l e - " > - 4 e ( 1 + 2sinh(u)).
Dividing by (K2 o f ) and using K2 o f <_ K1 and K2 o f < - a , we obtain
eu - e-" < eu
gl
e_~ <
4e(1 + 2sinh(u)) < 4e(1 + 2sinh(u))
K2of
-
Whence
4e>2
(
l+2sinh(u)]
K2of
->2
-
l+2sinh(T/)]
a
"
This last inequality contradicts our choice of e. This contradiction implies
either t h a t :P is e m p t y or t h a t u attains its m a x i m u m .
In all cases, we have proved that u < 0 on S. []
255
MAFtC TFtOYANOV
REFERENCES
[1] Ahlfors L.V., An r
o//Schwarz's Lemma, Trans. Arner. Math. Soc. 43 (1938),
359-364
[2] Hulin, D. et Troyanov, M., Sur la cottr~ure des surfaces ouvertea, C.Ft.Acad.Sci. Paris
S~rie 1 310 (1990), 203-206
[3] Hulin, D. et Troyanov, M., Prescribing curvature on open surfaces, Pr~print Ecole Polytechnique, Palaiseau (1990)
[4] Mok N. & Yau S.T., Completeness o//the K~hler-Einatein Metric on Bounded Domain
and the Characterization of Domains of Holomorphy by Curvature Conditions, The
Mathematical Heritage of Henri Polncar~, part I Proc. Sympoeia Pure Math. 39 (1983),
pp. 41-60
[5] O~erman Ft., On the inefuality A u >_ //(u), Pacific J. Math V I I (1957), 1641-1647
[6] Yau S.T., H~=rmonic//unctions on complete Riemannian Manifolds, Comm. Pure Appl.
Math. 28 (1975), 201-228
[7] Yau S.T., A generctl Schtv=rz Lcmma ~~or Kahlcr Manifolds., Amer. Jour. Math. 100
(1978), 197-203
Marc Troyanov
D6paxtement de Math6rnatiques
Univerait6 du Quebec ~ Montreal
C.P. 8888, Succurs,de A
Montreal (Quebec) H3C 3 P 8 , Canada
(Received December 18, 1990;
in r e v i s e d form May 7, 1991)
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