two points is defined on R, on the Alexandroff one- point compactification of R and on the .... Then (Theorem 7) it is shown that d = X - X , dKD = X â A* and dHf, = X*-X', ... ÐЯ*du. = 0, while u = iQ)L ,u implies that u is constant on each contour of Я and ... say Яy Я2,-- - ,Яm. Each Я. is called a part of yy Thus, {aQ, a y yQ, Я v.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 171, September
1972
EXTREMAL LENGTH AND HARMONICFUNCTIONS ON RIEMANNSURFACES BY
CARL DAVID MINDAÍ1) ABSTRACT. Expressions for several conformally invariant pseudometrics on a Riemann surface R are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on 7? having finite Dirichlet integral. The reduced extremal distance between two points is defined on R, on the Alexandroff onepoint compactification of R and on the Kerekjarto-Stoilow compactification of 7?. These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.
Introduction. Riemann
surface
Ahlfors duced
Three
R. These
and Beurling extremal
removing length then
are related
[2] and to their
distance
disks
of radii
of the family adding
new forms of reduced
A(£., r
ÇA between
of all curves
the factor
point
except
the curves
compactification
tively. tions
that
These having
prescribed
key to establishing (Theorem
R
reduced
on 7? joining
1) dealing
connection
with extremal
defined
For example,
by
the re-
Ç,., t,2 on R is defined considering
the boundaries to obtain
the disks
by
the extremal
of these
a quantity
are taken
and the Kerekjarto-Stoilow distances
singularities
this
name
for a
disks
with
and
a finite
C2) and A*(£1? £2) are defined in a similar
between
extremal
[l].
two points
(1/277) log ir y*2) in order
are defined
of the same
of perimeter
zz e HDÍR)
Let
with
of u £ HD(R) 0. A cycle
is a cycle
sures,
that
the integral
in R ~ E homologous
which
provides
pseudometrics.
classification
theory
invariant
pseudometric
this
of Riemann
d^M,
of * du over every if for every
dividing
in the Hilbert
length
is applied
surfaces.
interpretation
to derive
tests
A Riemann
of the various
surface
and only
if the extremal
distance
8, 9) that
between
R £ 0„D
any two compact
or R*, R'
contains
(0„D,
subsets
or R*).
in the
R is of class
(Theorems
on R or R'iR
mea-
for degeneracy
functions.
computed
E C R
HD(R).
constant
whether
cycle
set
of harmonic space
HD(R), KD(R), HM(R), respectively,
same
ate defined
compact
0HD, 0KD, 0HM if the class It is proved
over all
= X - X , dKD = X —A* and dHf, =
extremal
result
and
of
u on R
of |zz(z^j) - zz(£2)|
dKD
dividing
of KD(R)
a geometric
Finally,
from classes
functions
to y. HM(R), the space
complement
which
KDÍR) and HMÍR) of HD(R). KD(R) is
Then (Theorem 7) it is shown that d X*-X',
are defined
pseudometrics,
the subclasses such
surfaces
harmonic
the supremum
y on R is called
is the orthogonal
of all
D(zz). A conformally
D(zz) < 1. Two other
the subset
pseudometrics
on R by taking
by employing
bordered
[4].
be the class
integral
y on R is
there
invariant
HDÍR)
is introduced
analogously
and Rodin
conformally
Dirichlet
£2)
2, 3, 4) to noncompact
of Marden
functions.
[September
only
0„M)
if
of R is the
The result
for 0„D
is
due to Rodin [6]. Finally,
several
a conformally
explicit
invariant
computations
metric
are given
of the reduced for hyperbolic
extremal
simply
distances
connected
and
Riemann
surfaces.
1. Preliminaries.
For general
results
about
Riemann
surfaces
see
Ahlfors-
Sario [3]. (1.1)
At times
harmonic
we shall
function
borhood
having
of the ideal
low from the main
3A, p. 154],
boundary
[8, Theorem
L 0, il)L j and
cise
meaning
vanishing
derivative
+ uydy
on R- Likewise,
¡ß*du
= 0, while
A knowledge
of a harmonic
is the star
u = (/)L u near
each
contour
hold in an appropriate
that
conjugate ß means
that
These
function
usually
normal
operators on ß.
fol-
([3, Theorem
opera-
give
pre-
For example,
of ß is akin
to having
is, * du = 0 along
the boundary
of the differential
form
that
u is constant
u is constant
of ß vanishes. limiting
will
of a in a neigh-
operators
of the particular
[8, pp. 46—51].
behavior
uniqueness behavior
R. This
of normal
zz = LQu in a neighborhood
u = iQ)L ,u implies
of * du over only
surface
of the theory
on the boundary;
* du = - u dx + u^dy
statements
of a Riemann
2B, p. 4l]).
on R, then
and perhaps
and prescribed
iQ)L x is necessary
to the boundary
normal
the existence
singularities
theorem
if zz is harmonic
the integral
ß
existence
tors
where
require
certain
sense
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
on each Of course,
on a general
a
du = u dx
on ß and
contour both
of ß and
of these
Riemann
surface,
1972]
EXTREMAL LENGTH, HARMONIC FUNCTIONS, RIEMANN SURFACES
but they
are to be taken
that
the Dirichlet
literally
integral
on a compact
of a harmonic
bordered
function
DR(u) = D(u) = jj
Riemann
3
surface.
u on a Riemann
Recall
surface
R is
(u2x + u2)dxdy.
R
The
subscript
R is frequently
omitted
when
one fixed
Riemann
surface
is under
consideration. (1.2)
The notion
will be employed.
rectifiable
A locally and
rectifiable
to be locally
for a family
of this
topic
i]. A curve
closed
or half-open
if the restriction
rectifiable
rectifiable.
rectifiable
over all compact
curve.
Let
curve.
of chains
see
Rodin-Sario
subpaths
[8, pp.
on R is a continuous interval
on the real
all curves
is defined
line.
density
rectifiable
c : I —>
The
curve
c
of / is rectifiable. 72. is an integer
and chains
are assumed
on R and let
to be the supremum
of c. For a locally
121 —132]
mapping
formal sum c = S 72.c. where
Henceforth,
ds = p(^)| i2))function
to be continued
around
In this
tegral
of * du = du* over
ß.
after
once.
ß.
line as a continuous The function
real
part
nate
words,
D~(u) = Í
J dR dR
we see
that
u* decreases
l0(s)
subsets,
C will be finite
For each
u increases
Jordan
the connected
level
tees
that
direction
of 7Q(s). denote
Ç(s)
or yv Note
cause £'(t)
the first
that
c(s)
the path
the integral
for which
of * du over
if and only if £(s)
If not, then the level
open
J», -,
neigh-
of any local
subset
derivative
*du = -f
coordi-
of the upper
and lm{f (z)\ < 0 on
u* is strictly
decreasing.
From
* du,
Ja -u0
= £(t).
curve
C, of the interval
I As)
direction.
it passes
through
through
any critical
put
£ ?.
Otherwise,
lQ(s)
stops
along u*(£'(s)) /3. is Ii £'is)
ß..
The function
Continue
= u*(C(s)). 0. For future
Iy(s) of u* which
point
originates
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
which point
point
of
of
of u, then
on some about
ß.
it
guaran-
in the positive
ß.
until £'(s)
increas-
arriving exists
we note
that
of u, then
put
at ('(s)
c(s)
ß. C y
I Js)
u* is initially
u* along reference
Orient
s in J and write
Such a point
is a critical
be
in J.
a critical
points
on ß ■; our assumption
of u. Now move along
shall
along
a critical
on a,,
by b..
a..
is a curve
terminates
point
O]
a chain
of u* leaves I As)
passes
not cross
(— D(u),
associate
lQ(s)
determined
u* from £(s)
point
in terms
so that
lQ(s)
of lQ(s)
is not a critical
ing as we continue C'(s),
When
the endpoint
following
to the real
f = u + iu* has nonnegative
imaginary a_
arc unless
curve
al
£(s)
to its initial
be extended
0 < u < 1 on R, in a relative
curve
s in C. If 7Q(s) does
Let
are
the in-
from 0 to — D(u) on [O, l) provided
level
77 in 7?, place
in place
u*(aQ(t))
Functions
The fact that
u* returns may
s £ S we shall
ends
on either
that
each
in the positive
* du = 0 and is an analytic u. In case
u* harmoni-
on a0.
fashion.
Therefore
u*du=\
5 and
and with
s e ( - D(u), 0] a single
so that
77* for a in a neigh-
from 0 to — D(u).
Two complementary defined.
u* to the curve
u* to all of aQ so that
onto a relatively
f (z) has a purely
contin-
aQ([0, tyY). The value
function
on a„.
du* = * du < 0 along
Suppose
1.
= 0. Since
a neighborhood
the function
In other
period
the holomorphic
imagininary
such
for aQ. harmonic
Extend
u*(b.(t))
decreases
z2*(a.(o))
of a.
and is purely
z mapping
half-plane, aQ.
point
having
this
u* is continuous
guarantees
of
Through
function
we extend
Explicitly,
conjugate
and then continue
73. in a similar "7
strictly
so that
of each
manner
vanishes
function u*(aAt))
u* is normalized
a conjugate
= u*(aQ(t~))
a, 1 and the
traversing
the curve
on [0, l); in short,
value
a0(0).
to 22*(aQ(0)).
u^ia^tj))
a harmonic
We illustrate near
along
determine
that
is a continuous
borhood
manner.
but need not be equal
of a0(tA
in continuing
of u defined
u* may be defined
exists
tA)
borhood
are used
5
moves
at be-
£'(s)
=
s in C.
into
R
6
C. D. MINDA
when oriented
ß.
neat
Çis).
in the direction
Ç'is)
since
Thus,
ation
-> c.]7
of lQ(s).
u is strictly
either
ax
and put
* du > 0 along
zz + zzz* maps
in \z; Relzi
of increasing
where
If I As)
values
by £(s);
of u. In fact,
this
neighborhood
also
* du < 0 along
shows
of Ç is)
that
C'is)
4
onto a relative
open set
u = c.1 on ß.. The level curve I 1As) is called a continu~i a critical point of zz, then 5 goes in C. Otherwise,
crosses
increasing
or some
ß.
a relative
[September
on I As)
so that the connected
ß,
with
k 4 j. In the former
s in S. Again
c(s)
£ J.
In the latter
level
case,
curve
write
we repeat
I As)
ends
on
c(s) = ¡0(s) + Ix(s)
the construction
given
above. In this every curves
J.
manner
we see
s £ S there
of zz* obtained
For
then
that
is a finite
I (s) = l,(t)
Otherwise, so that
because
through
and
/,,(/).
Continuing
is finite
because
R and through Finally critical
they
,(/)
each
we establish
points
can be paved
¡k(t)
manner
point
with
For any linear
little
density
so that
over
X(j)
3. Extensions izations surface
of Theorem having
ate partitioned
number
parameter
curves
C
surface
of
and
of zz and
R
u*.
p2du\A. /
that * du = 0, du > 0 along ic(s): s e S\ C J,
pduY < f '
D-(u)L2(A, ds)
ß if for every
R
[8, Theorem
of Theorem
the identity
to the same
convergent
compact
partition;
R'.
set
2B, p. 58].
1 is the analog in other
be the Alexandroff
to R to obtain
on compact
Given
of fi to
A(3) = D(z2 )_1. in which
all contours
of y,
compactification
a sequence
E C R there
Hence,
of the case
words,
one-point
subsets
belong
of R and
{£ ! of points
is an integer
y„ =
N = N(E)
in
R, write
with
the prop-
erty that Cn d E for all n > N. Similarly,
if 7: [O, l) —» fi is a curve, then /(/) —»
ß as
set
/ —►1 means
that
for each
compact
l(t) 4 E for 1 — S < t < 1. Say that sequence
ing
t
aQ to a y A(j
stricting related
each
either
7 is a curve
family chains
curve
Cj
with of y
J . Both
y
are equivalent
and
consisting
a„ K
families
J
X
and
from
contain
of the family
jj
ciQ to a;
/.(/•) clusters
J.
The following
for extremal
of chains
a.
to a
/ —►1. If we define
natural
length
that
and
result
shows
problems.
Proposition 1. A(3') = A(§') = A(K').
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
by re-
< A(3). 7 such
Two that
I.: [O, l) —» R is
H'
together 7.(l) = ß
/„ + 7. may
way we regard
connect-
obtained
A(3')
/ —►1 (/ = 0, l).
in R joining
In this
that
or 7= 7Q + 7j where
on ß as
in fi'
of chains
is the family
a . to /3 in R
to a
of curves
3"' D 3", it follows
are needed,
Z.0) —> ß as
is a 5 > 0 such
on ß as t —» 1 if there is a
be the family
length
to R. Since
of the curves
that
a curve
joining and
Let
in R joining
/n + Z, such
in R
) —» ß.
in ?'
7.(o) e a.
7. becomes
l(t
) is the extremal
curve
families
a curve
then
—> 1 with
EC fi there
l(t) clusters
is the subwith the
in this
be viewed K'
that
case,
as a
as a subset
of
3',
K'
§'
and
8
C. D. MINDA Proof.
The inclusion
Jv
C j'
implies
[September X(i\ ) > A(j').
Every
curve
contains a chain in K' so that A(3"') > A(K') [9, p. 321]. Therefore, In order
to establish
A(K') = A(L 0 be any number
that
result
of R and set that
lim A(Lf ) exists.
to demonstrate
ds)AiR,
into
we use this
I such
lim MJV ) < À(Lf').
to show
is divided
is a curve
exhaustion
chains
[9, p. 321], and hence in \j,
C/
of R, and then
72
the collection
to a
where
The proof
that
1972]
EXTREMAL LENGTH, HARMONIC FUNCTIONS, RIEMANN SURFACES
A(R, ds) > 0. Let
a
= L(\-l , ds), then
a2tí—< A(Rn , ds)\(§'" so that shall
a = lim a
prove
there
n
{a \ is an increasing
)< A(R, ds)\(§'" —
is an I £ §'
such
that
with
in fi
L(7, ds) < a + 5c. Then
the existence
joining
a.
¡n = lQn + lln where
denote
the first
00. This
is possible
that the points
C-ik) £ y,
also
because
Let
for every
V ik)
is still
we again
ds) < a +
that in passing
which
Otherwise
for k 2(0)\.
C2- A*(£,,
£2)
to the Kerékjártó-Sto'ilow
one-point
1 2
is established compactification
is the reduced
extremal
compactification.
exactly reduced
distance
Because
3(7y r2) C f'(rv r2),3*(71?r2), it follows that \(£y Q > A'(£,, Q, A*«,, ¿2). Also, if R is compact, then 3(7^ r2) = ¡Piry, r?) = 'J*(rv r2) and A(£,, C2) = A'(C,> 0 we must show
local
coordinate
in the complex
of d„,
explicitly.
plane.
z.
This
that
means
Employing
t?
< dH(V)(^0^1)
e V" and Zj = z(C,y) £ U. It is possible Let
u £ HD(U),
then
there
there
is a neighborhood
that
z(¿,A) = 0 and
the monotonicity
N
disk
z(V) = U
and conformai
inequality,
* dH(U){°' Z
to compute
is a holomorphic
the real part of which is u. If f(z) = 2~_0 the Cauchy-Schwarz
£ R.
in
we obtain
dH(R)^0'
Í,
£
C2) < dH(Ry^V *=2^*s lmmediate from the definition of dH.
C0 £ R and
of C0 with the property that N C \Ç: dH(£Q, £) < c\. Let V be a parametric
where
is
a^z", then
d^.yAo,
function
/ defined
D(u) = 272~
we have
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
zx) in
Í7
7z|a | 2. Using
16
C. D. MINDA
[September
l^j) - «(0)|2< l/U^ - /(0)12= ¡¿a^
W=i
CXJ
»
I
|2
,
^"K^Z^-^iogd-iz^)-! n —\
n— 1
and equality
holds if 77(z) = log |l - z ,z|_1.
[(I/77) log(l
- |z1|2)~1]M.
N = i£: |z(£)| (v) Since
functions
Let
d
- u(ÇA\,
Establishing
a be the upper
dH^V
where
upper
limit
¿H
of the family
dH is certainly
semicontinuity
requires (¿\,
a normal
)| < 1 for every
that
Ía
(vi)
72 so that
S is uniformly
The supremum
the supremum
Iu(£)
- u(£A\
induced
by d„
{u lis
is
coarser
is subharmonic
classes
In the same
on compact
than
that
a