extremal length and harmonic functions on riemann surfaces

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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 — A* and dHf, = X*-X', ... ЎЯ*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

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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

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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

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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