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Generalizations to space of the Cauchy and Morera theorems

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GENERALIZATIONS TO SPACE OF THE CAUCHY AND
MORERA THEOREMS
Maxwell Reade
A thesis presented to the faculty of The Rice
Institute in partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
June 1S40.
Hf-¥T93
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UMI Number: 3079848
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GENERALIZATIONS TO SPACE OF THE CAUCHY AND
MORERA THEOREMS
INTRODUCTION.
Consider the function
CD
iipa
- Xi CU,0-}1 XVCtblhh
defined in a simply connected domain D
.*
A neces­
sary and sufficient condition that -ft*) be analytic in
f) is that the Cauchy-Riemann equations be satisfied
there x1
-
K
Sir
*
-
ttr
2Ky >
2)u
that is
(2 )
^ur-o,
where
is a differential operator.
From (2) we obtain
( 3 >
which can be written in the form
( 4 )
E
j=/
Conversely,
j
T
^
'
r
3
”
°
•
=t
(3) implies either (2) or
A ur-o ,
*We shall consider only finite simply connected do*
mains.
1.
See the theorem of Looman-Menchoffs
S. Sales,
Theory of the Integral, sec. rev. ed., (1937), p. 199.
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-2where
7 _ *5
• 7)
A = 'ZCL i tlT
CTf '
If (3) holds, then we say that
and
form a couple of con.iugate harmonic functions.
The real functions
(5)
Xj-z.
defined and continuous in a simply connected domain
D
, will he said to define a surface
S
.
if the
first partial derivatives of the functions (5) are
continuous and satisfy
(6 )
PCU,U^« CrCU.U^
PCtWl^s-O,
where
■4 - r if*>\x
^ = 2 1 ^
S-\
(7 )
F=
Sif-*’
Z_. ^u. ^ ‘
J _l
(La T" (£fj
' r~ bo- *
*
are the coefficients of the first fundamental dif­
ferential quadratic form,1"
c/s*"- EduJ'*- ZFjiudif 4 Grd<f‘%'
of the surface S
, then the parameters «,ir
are said
3
to he isothermic parameters, and the surface is said
to he given in isothermic representation.
P
on 5
is conformal except where
The map of
.
Now (6 ),
which is a generalization of (4), can he written in the
form
(8 )
JLCA
j-i
O*
2.
W. C. Graustein, Piffe-rantial Geometry, p. 82.
3.
Graustein, op. cit., p. 131.
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-3which is a generalization of (3).
The following is a combination of the Cauchy and
tj
Morera theorems.
If the function 4-c.fc} in (1)
simply connected domain
continuous in a
, then a necessary and suf­
ficient condition that ^
analytic ip D
is that
for each closed rectifiable Jordan curve ft
in
D
lying
,
(9 )
^ k w ^ - o .
We may write (9), when we separate its left-hand
member into its real and imginary parts, in a form
analogous to the Cauchy-Riemann equations:
But the relations (10) imply
j'-i
j =l
w
which are analogous to the relations (4);
and (11)
may be written in the following form analogous to (3) s
(12)
t\ L
S-i
*
A generalization of (12), analogous to (8 ), is
(13)
Z [ k <;tu',rt‘ii f = o ,
j-i
4.
E. J. Tovmsend, Functions of a Complex Variable.
p. 82.
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-4where £
is an arbitrary closed rectifiable Jordan
curve lying in
f) ,
In this paper we shall study (13)
and
J—I
where
Kyiuuirtifcj - fin.*) ,
^L
is the circle in J) with variable radius A
and fixed center Clu,^
, and where
is a quan­
tity (not always the same quantity) such that
A—■>O
A.x
/u
~
If the functions (5) are harmonic and satisfy (6 )
In a simply connected domain b
, they have been called
s
a triple of conjugate harmonic functions. In terms of
this definition, a theorem of tfeierstrass^may be
stated as follows.
A necessary and sufficient condition that the func­
tions (5), defined in a simply connected domain, be
the coordinate functions of a minimal surface given
in isothermic representation is that they form a triple
of conjugate harmonic functions.
We recall the following theorem.
Theorem A.
5.
i
A necessary and sufficient condition
S. F. Beckenbach and T. Radd, Subharmonic Func­
tions and Minimal Surfaces. Trans. Am. Math. Soc.,
v. 35 (1S33), pp. 648-661.
6 . Loc» ext., p. 649.
7.
Graustein, op. cit., p. 98.
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-5that the functions (1 ), having continuous partial de­
rivatives of the third order in a simply connected do­
main. he the coordinate functions of a spherical sur­
face
that there exist a constant
06
such that
where
e _
\
-
J fin
Xavrl
»
i* lu u - X-L.U- - f W I
^
j^ J
J- —---------- ,
H
H
H
are the coefficients of the second fundamental quadratic form of 5
t
, and where
V e f r - F1- ,
^
'
Here we have used the notation
I Q-v.
=■
Ou <W 0.*
bi k ^
\ C.v
C. C*
It follows immediately that a necessary and suf­
ficient condition that the functions (5) be the coor­
dinate functions of a spherical surface given in iso­
thermic representation is that both (6 ) and
(14)
e' ~ l >
hold.
If the functions (5) have continuous partial deri­
vatives of the
main 0
awv. th order in a simply connected do­
, then about each point ftia.vTo) of 0
we have a
finite Taylor expansion for each functions
S.
If
q
, then
'S
is a plane surface; loc.
cit., p. 97.
9.
Loc. cit., pp. 93-94.
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where
a - U o - a toi © >
iT-tro -fi
*
where
t^O- 2 - -f
£cr
is a differential operator, and where the partial de­
rivatives are evaluated at
Ctu, tf6) .
We establish the following result for functions (5)
with Taylor expansions (15).
If C/v is. the circle in D with center at c«.o, if
radius
a.
, then
IC
(16)
H[J
j‘-_j
lls?'*1'
KTa-O ^)=o
CA
where
^
t+i
^
* "V;.
-fe-
where
0tt*' air-.’V
where
* r
^
,c
•
’
-
/
3t dC-sl'
and where
ifi the greatest integer not greater
than <,(IV*—
Proof.
We make use of the following equalities.
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-7.
,
, r t ^ ) r c ^ j ,
PM
Ar
/A - + ----------------t&i?&
de-^i
AV\+Wf-1_\
2_
TT
/-.7 \
''
»
-
V
C
Jfc
'
1L
( SuA^fi C.O&*© J.&- - f
Jo
Jd
+
/
C-i^(0sw*0 cos-4© dd t
( ^ 9 m se i© ^ [ i t - C - O ^ U ^
Jo
o
0 <***« ^
Prom (17) we obtain
'sux*® e0-s»M<9ci© = I?/aaja<*. f
(18)
(b)
- o
where (a) holds for v*
holds for
ma
and m
both even and where (b)
and At not both even, and where
r t N g j r t ^
p^ w f u f l .
a.
=- Aj
'
r »1------------W > r(s*i>
---
,
Ojljl,' ■
rcf^+o
where w^j.^ and
% .
But-
10.
W. F. Osgood, Advanced Calculus. p. 485, ex. 2.
11.
These results follow from the following pro­
perties of the Gamma function (loc. cit., pp. 482-483)s
T C r o d ,
for O o
,
rcO-a-i
and
rc^=Jir.
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-8therefore (18), (a), may be written as
(19)
(
vx '
Jo
-311—
fcffK O p ,
7*
? ' <£•' CP + P '
*
-f’i "
.
Consider
where the partial derivatives are evaluated at (u0>cr0) .
We obtain
(2° ) f | ( w e x
siii l H - Js -0; =.4V ''
^ 0o
Here we have used (18), (b).
Similarly we obtain
(21)
J \(co*©£- -KtMa-^) /f/lsoiS de -o,
o *>
to
dt*-' -i
Consider the expression
( 22)
^^
* =<Vj' '
j -ljz.j3.
Applying (18), (b), to (22) we obtain
f^ T -
1 ***
s
&
« ]« * /»
% ,£ % _ je
which, by (19), may be written in the form
LIT
_
_ 3-=,tl ,
. s
i o
\C
e g & o
-t-tux©
^
S K: hm£>
=
m t O /
JiS *\I
^.j
L
* L _
1
sTa^oi &Uir^u'w t ]
(23)
H
^ l l l L 2 -a.V
2.15 S( C£+/i' So. j ->
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J
.
-9Similarly we obtain
3 = -0 ,
(24)
'ixs 5{«-t-or a#-
J*
From (20), (21), (23) and (24) we obtain, for the
functions (15),
1*4-1 1
U;iu,^At-rr^«r^r
from which we obtain (16).
For
, (16) is displayed in (26), (72)
and (58) respectively.
1.
CHARACTERIZATION' OF ISOTHERMIC MAPS.
Theorem 1.
If the functions (1) have continuous
partial derivatives of the first order in a sinrolv
connected domain
0
, then a necessary and sufficient
condition that they map 0
S
ig, that for each point
(25)
isothermicallv on a surface
CMo,^)
D ,
t i l
*
.
j- i
where Cv
0
jj=[ the circle in
and radius
Proof.
with center at CUa>fo)
A
If the first partial derivatives of the
functions (5) are continuous in j) , then we obtain a
finite Taylor expansion for each function about an
arbitrary point C(u,ir0) of j)
If
C/l
is the circle in
0
by setting /*v- (
in (15).
with center at C U o and
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-1 0 -
radius
a.
, then upon setting
in (16) we ob­
tain
(26)
From (26) it follows that a necessary and sufficient
condition that (8 ) hold in 0
is that (26) hold.
This completes the proof of Theorem 1.
2.
CHARACTERIZATION OF THOSE ISOTHERMIC SPHERI­
CAL MAPS THAT DO NOT MAP CIRCLES ON CIRCLES.
Lemma i.
If the functions (5) are not identical-
IX constant in a simply connected domain D
they map P
on a sphere
is_othermically on a surface
J
Qf finite radius
(X
, and if
3> that lies
, such that
circles are mapped on circles, then the functions (5 )
have the representation
where the
are real constants. where
has one
of the following forms.
frirtf
1
frt+S
and where & $1%) ia the real part, and
the imaginary
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part, of
.
Proof.
Since the functions (5) are not identical­
ly constant in D
sphere.
he
, it follows that ^
is not a point -
Let the coordinates of the center
CO-i* d*., 0.-*) .
Q
of $
Since the functions (5) map D
thermically on 5
iso-
such that circles are mapped on cir­
cles, the stereographic projection of aJ
on the
-
plane, the coordinates of the pole of projection be­
ing (CL•>ft A<-) j
where X
and /-^I
mum of the two quantities ICU*&I
duces an isothermic map of
D
is the maxi­
and
I
on a domain
, in-
D
of the
$,■6 -plane whereby circles are mapped on circles.
Here the
3,-6
positive
S - and
Xi - and
-plane is so placed and oriented that the
& -axes coincide with the positive
X^ -axes respectively.
^
P
on D
The function mapping
I^
must be linear, and therefore of the form
Ca)
=
S b+S
(6)
+
ft* S'
where
(a)
holds if the map of
P
on
P
is directly con-
formal and where (b) holds if the map of
inversely conformal.
D*
P
on P
is
The stereographic projection of
is given by
12.
This follows immediately from Theorem 25,
p. 32, of L. R. Ford, Automorphic Functions.
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12
^l Ou |Jtl (S-tf.i)
C s-ao1-*
Q v+
(27)
*«^l
« ^ > -----
c s-aY’
+tt-flo'vlv
<3.(2.li-l
*•
t+Lir-Qj'+X1' ■
CS-CU
It follows that the map of D
- On
on S
is given by
litI <R ft-*-)
------------
*
(28)
where
Vur)~ P-ifc)- t A . + c a J ,
and hence it follows that the functions (5) have the
required form.
We note that if it is given only that one non­
J)
null circle in
is mapped on
a
circle on
S
by
the isothermic functions (5), then, as in the above
discussion,
O
is mapped isothermically on
such that one non-null circle in P
circle in
linear;
D
.
D
is mapped on a
Therefore the mapping function is
hence it follows that the functions (5)
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-13have the representation (28).
Lemma 2.
If the functions (5) are not identical­
ly constant in a simply connected domain
0
, and if
they man P
S
that lies
isothennically on a surface
&ZL a gffhere J
gf finite radius
0. , then a necessary
and sufficient condition that they man circles on cir­
cles
that the first quadratic form of
5
have the
representation
(29)
cU*«.
Uu-uo^ur-u^ + b* \
Necessity.
If the center Q
l>o,
of d
is at
then, by Lemma 1, the functions (5) are given by (28).
From (27), a computation shows that we obtain
£
H e l"
=
X- >
I*
where
m
$ Z
( £')
.
J =1
But
lt
t /*r-ptr
l^'COI *
— T
1
and
Cs-atf-w* -a*y- - 1kttl"--1
^ ;
therefore
13.
Compare with L. P. Eisenhart, A Treatise on
thg Differential Geometry of Curves and Surfaces, p. 109.
14.
Compare with Ford, op. cit., p. 118.
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li<t+piVJl'-|C«.+Av ] *■
which may he written as follows:
(30)
0^
E -
[tu.'U0)u+(.iMrfl)v+ ^ \ x
where
-K a o -
-iUo = c^ifw-^,)+iv((rir1. - ^ <r,)J
bv -
(f|V+ A M l T , CttflT-nT^,
N
H<sl'u£-wr
f\ -
where the
^
>Q%
ifcV U l V > &>
, fir;
and Fj are real constants, such
that
oc - a^-tt aTi.,
^ -i
+ c ^v,
<Sr"=- 'fi■+<■ Tv-j
S'
S, 4 * ' JL
,
Here we have used the representation (a) on page 11
for Ftfc)
in order to derive the relation (30).
Simi­
larly, we can derive (30) when F64-) has the form (b)
on page 11 , the values of the constants (T0
and
b*~
in (30) being altered accordingly.
it
The Gaussian (total) curvature of o
(31)
K - ^
15.
ze
is given by
J
E. F. Beckenbach and T. Rado, Subharmonic Func-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-15which, with (30), yields
k -
'
Cv
But jb
fore
is a sphere of finite non-null radius;
>o .
Under the hypotheses,
meters.
u>iT
are isothermic para­
From (6 ) and (30) it follows that the first
quadratic form of S
Sufficiency.
D
there­
is precisely (29).
Let the stereographic projection of
on the sphere
be
with center at
, where ^
b-
is the sphere
and radius
and where the pole of projection is at
(.CUtTo, b)
.
This projection is given by
CtU-Uo^
* v- - U o + - ----------------------------
CU-Uo^V CVT-UaiV bV
^0+
(32)
^
^\}
>
Ca-.u.oit'+ctr-tr0)V(>w I
_
The first quadratic form of
o
*
is found to be
(33)
\ (U -U o ) V
ClT-{To)1' + b X
which is identical with that of
S
1
. It
follows
If-
from (29), (31) and (33) that
5
and
S
tions and Surfaces of Negative Curvature.
lie on
Trans. Am.
Math. Soc., v. 35 (1933), pp. 662-674.
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-16spheres with equal radii.
We shall show that to each circle in D
there
corresponds, by means of the functions (5), a circle
on
S' .
Consider the non-null circle
and let the interior of Q
functions (3A) map R
on
yf
S
be denoted by
R
is the image of
map R
on a surface H
C
^
,
•
The
isothermically on a surface 27 ^
which is bounded by a non-null circle
where
curve
in
Co
on S
.According to the
C
"if,
, and the functions
(5)
which is bounded by
a
reasoning in the proof
/c
of Lemma 1,
it is sufficient that we take the non-
null circle Co so\small that the spherical cap 21
■jp
is contained
on a hemisphere.
*
There is a pole of
i S'* .
Let Pi*
be a
point of £
, and let Pa* P* be the arc of the
n
p ^
q ^
geodesic passing through ra and
and lying on
jf.
^
.
Let the image of P»
image of P/ in D
P/
of
in D
on S
be
-
in P be
V 0 » the
and the image of the arc
be the arc
. Let the image
be Po , the image of <£>,on
S
be
P,
16.
See the bottom of page 12.
17.
The geodesics on spherical surfaces are arcs
of great circles; Graustein, op. cit., p. 149.
is on a hemisphere; hence through each pair of points
-4c
there passes exactly one geodesic that lies on 2 -
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•
-17and the image of
on 5 be P*P/ . Now f<sP,
t*
is a geodesic. Keep "*> fixed and allow r, to de'*r~^
scribe the circle
C
.
is of fixed length ^
^
^
As P/ describes £
The geodesic arc
for all points
P'
^
Pa P>
on
^
.
Q
,
•
From (29)
and (33) it follows that the geodesic arc
also
m
describes
is of the same fixed length ^ for all points P»
C
.
Moreover, as
P»
describes d
arc
?&
in
An
on S
describes ^ exactly once around,
exactly once around.
D
For, let the
be the image of the geodesic arc
and let the image of
the geodesic arc
on S
, so that P6 ^
are corresponding geodesic
18.
are on C
arcs. Let P&*A *
and A» and 4v are
is a geodesic arc.
and
geodesic arc P*P/
corresponding curve
on
C
Here
. Since
For, the geodesic
arcs are arcs of shortest distance.
not a geodesic arc on 21
-be
and P6^'
be another pair of corresponding geodesic arcs.
Ai and
on
If
PoP,
were
i then there would be a
through Po and Pi on Z7
P#*Pf*
would be of the same length.
through P» and P'
Since P» ft
are of equal length and since. P»Pi
and
.
The
on
Pa* P'*
is shorter than
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-18the functions (5) and (32) are isothermic, it fol­
lows that the angle formed by
and
a,
is equl to that between 11> A/
and
a H
P<a Aw .
Hence there is a one-to-one correspondence between the
points of C
and ^
Hence the curve
at ?0 •
Co is
C
as P/
on
S
describes C
a
is
once around.
circle with center
Hence we have shown that the non-null circle
mapped on a circle
C
on
S
by the functions (5)
Hence the functions (5) map all circles in
therm! cally on circles on
Theorem 2.
^ iso­
S
If the functions (5) are not identi­
cally constant in a simply connected domain
0
,
and if they have continuous partial derivatives of
the third order in D
, then 3 necessary and suffi­
cient condition that they map D
a surface ^
isotherm!callv on
that lies on a sphere of finite radius.
such that circles are not mapped on circles, is that
for each point
vUa.iTo) in D
j
ii.
(34)
where Ca. is, the circle in 0
and radius
ji
, and
that there exist a point CUi>(/Y)
it follows that
This is not so;
with center at
hence
ft
is not a geodesic arc,
is a geodesic arc.
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in
D
such that
«6)
r \ I ^ X/CU,fF)
-4(ftft*)i
J -I
where
is the circle in j) with center at
and radius jv. .
Necessity.
map D
Under the hypotheses, the functions (5)
isothermically on a e^Qface S
that lies on a
sphere of finite non-null radius.
Hence (6 ) and (14)
hold, and there exists a constant
*
(36)
e
^
of.F,
For the present representation of
of Gauss
<?
such that
gs.
S
, the formulas
become
*>uu “ R u-XjY - RirXjV- f>e
2. Ru“Xj'a 4- Ru. x ^
Xjwr- -
where
and where
(38)
in cyclic order,
are the direction cosines of the normal to S
We shall need the following relations.
19.
Graustein, op. cit., pp. 135-137.
20.
Loc. cit., p. 92.
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.
-20-
4 A *;
-
(39)
*; ^
i-\
(40)
T inHif ^
j'il
, < * ( i f ) v,
To obtain (39) and (40), we shall use a method
which depends upon the existence and continuity of
the partial derivatives of order higher than three
of the functions (5).
These functions map
thermically on a spherical surface.
0
iso-
Therefore, as
in the proof of Lemma 1, we can consider an intermedi­
ate stereographic projection to show that the func­
tions (5) have the representation (28), where either
•?(.■£)
or
is analytic in D
.
From this rep­
resentation, it follows that the functions (5) have
continuouspartial derivatives of all orders.
To obtain (39) and (40), we shall need
the follow­
ing equalities;
(4 1 )
(42)
Ax; 5 a.(e
\
3
(43)
£ S ; < U ; =^o,
5-{
3
(44)
Z
A<jiC=0*
JV'l
3
(45 )
H
av£; /U; ^o,
Szi
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+
-213
(46)
21
j =i
A.X-i & Kj — & >
3
(47)
XL
J=(
AX] fcA-Tj -so.
It follows from (37) that
-xACe zj) >
and hence it follows that (41) holds.
Operating
on (41) with the operator
A
(48)
A5; -f-Ae AC; * ^
-=9.(e&?;
and operating on (48) with
Recall the equalities:
we obtain
A
we obtain (42).
%■i
(49)
(50)
j-I
(51)
71 Z;
t -\
i
i
(52)
j -*
i
(53)
+* - Z
3
= - H
KjV
,
We note that (43) and (51) are identical.
From (8) and (51) it follows that
X I Cu.
21.
- Z _ A;u ^ = Z.X-
Graustein, op. cit., pp. 93-94.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-ot
Hence the vectors whose direction components are
, <)V, < ; .
respectively, are mutually perpendicular.3'*' From (50)
and (53) we obtain
i
3
^ju. ~
j -i
j"-(
~
and hence the vectors whose direction components are
respectively, are mutually perpendicular.
(64)
ija
22.
23.
Hence
V ,
“
Loc. cit., p. 5.
Zju. and
.
K>‘u
are proportionals
*>a t
Hence from (7), (36) and (52) we obtain
s— oc .
Here f - o
only at isolated points.
For
if not, then there would exist a sequence of points
of
• • , and a point *© of &
following properties*
ao,
, with the
~
and
But, as has been noted on page
20, the functions (5) have the representation (28),
where either
or
i's analytic in
.
0
Then, as on page 14, we obtain
£r- _--- 4d.v Jta'
t
where
and
+ 1 1 ) 1
have the same meaning as in the
proof of Lemma 2.
hence
i W
/)Jc 4C>
-t*
'
,
.
Therefore -f'CW’o ,
It follows that -TCtlso
and
in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D
,
From (50) and (53) we obtain
YL
- T 1 <;*;u
J -I
J-l
and therefore the vectors whose direction components
are
respectively, are mutually perpendicular.
Hence
C55)
We can write (54) and (55) in the following forms
(56)
*.- 0Lkti ,
*46,
From (56) we obtain
£ Ax; A?; --«X<:Ax;>\
Jit
which, with (8), yields (44).
j's|
From (44) we obtain
A Z.A<j A£j ^ E
4=1
«f=t
/-<
*“
*
which, with (56), establishes (45).
From (44) we obtain
_ i>
XI1Aj(;K; - T I m J
/=<
i-(
from which it follows that -k*) is identically constant
in 0
.
But if foi) were identically constant in D
,
then it would follow from the representation (28) for
the functions (5) that the functions (5) are identi­
cally constant in D
•
From this contradiction of
the hypothesis, it follows that £
isolated points of D
24.
can vanish only at
.
See footnote 23.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-24which, with (56), establishes (46).
From (41), (43) and (44) we obtain
3
21 Aitj a Ax:; ~ o 3
i-1
which, with (56), yields (47).
From (42)-(47), inclusive, it follows that
(57)
H
A<i
- a A \ 21 AKy A 5 .
j =i
From (36), (56) and (57) we obtain (39).
From (8) and (56) we obtain
I
(A<j)v= o,
/-t
which combines with (36), (41), (49) and (50) to
yield (40).
Since the functions (5) have continuous partial
derivatives of all orders in 0
, the following ex­
pression, obtained from (16) by setting /vk - S' , is
valid:
(58)
ZL\ {
-- -TSV
v 3
-jr^5E l * Ax; a*A x;{-3C6Ax;)v]-e<r6 0 ,
«f-(
where the partial derivatives are evaluated at L(io,(Js)
and where Ca, is the circle in 0
and radius
with center at Cilicia)
is an arbitrary point in 0
Applying (8) and the relation
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.
to the relation (58),J we obtain
I
(59)
x;tu,(rtcU ^3L
J"-' cfc
™
Therefore (34) holds.
a A.k ;+*<*AX£>v1f<T(^#),
s-A
Let us now suppose that (35) does not hold, i.e.,
for each point (U.O(iT<>) in ^
,
(60)
,
j“-l
Ca
where C*, is the circle in D
and radius A. .
C6i)
with center at (UojU*)
From (59) and (60) it follows that
Yl[ t/ic; W x / t i c a / l x / H - °
M
holds.
From (39), (40) and (61) we obtain
(62)
But, by (36),
? an(i therefore (62) yields
(63)
~o,
From (63) we obtain
\e
r
which may be written in the following form:
2l A(
25.
^
A (Ijf
.
This relation follows from (47) and (56).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-26Hence
(64)
if'-f
e
)
where
(65)
\ &<.*) =-<D ,
Prom the imaginary part of (63) we obtain
Pvr
fr
F
which imply
( )
n
E a '- f
66
where
4«.^)
tion of tr .
e
4 it a )
j
£V - £
is a function of
cl
From (64), (65) and
e
4Uo)
j
and 4vtl>)
is a func­
(66) it follows that
the function
e
(67)
QLfc)
^ e
4du> ,
u
is an analytic function of t-=-u-t-tir
^
.
o)
From the
Cauchy-Riemann equations for the function (67) it fol­
lows that
£
where uto , ou
4du.)
=. ldoU--t-Gu ,
4vtu->
^ .
2S U o ^ +Av
and Q-v are real constants.
may be written in the form
which yields
(68)
^ci^-CKloU.-HLOdLuL,
e^ij.
gU-L
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
,
Hence (64)
-27From (68) we obtain
AoCuN:1^)^ouu-hk(a-ku,
-=.
where
is a real constant.
(69)
Hence
e -
y OloLU^^) +■ On.U-VdatT td>,|X
Substitution in (31) shows that
(70)
_
which, since
S
implies that
do^-0 .
is on
tjuobi ~ d
a
sphere of non-null radius,
Hence (69) yields
I
where
(71)
” dv
H «
*£O
From (70) and (71) it follows that
since
$
But
is on a sphere of finite non-null radius, it
follows that
form of
K ’s-Cfco'M .
5
.
Therefore the first quadratic
may be written as
JUS-
_
where
Cu>
Therefore the first quadratic form of 5
has the repre­
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-28sentation (29).
It follows that the functions (5)
map all circles in 0
on circles on 5
made use of Lemma 2.
This is a contradiction of our
hypothesis.
here we have
Therefore (35) holds.
Sufficiency.
By setting
*
j-l
;
-iv
in (16) we obtain
'
c*
*
“
where Q<\, is the circle in 0
/=(
with center at the arbi­
trary but fixed point CU&itfb) and radius A
, and where
the partial derivatives are evaluated at
.
From (34) and (72) it follows that (8) and
(73)
Z
A<j dJ*!
J-l
must hold.
Operating on (8) with \
, we obtain
j-i
Operating on (74) with the operator X , and applying
the relation (73) to the result, we obtain
(75)
X
& k; =-o .
j-i
The four real linear homogeneous equations in
HQ'
j^.i,
inulled by (74) and (7S) are
( 76)
i
Z Z .*ja
/=.t
*
/=<
4_
Atfj = lr — ^ KjutT
J~(
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4“
‘^iuir ^
j'-(
,
-29One solution of the system (76) is
(77)
which, by (8) and the theorem of Weierstrass, implies
that the functions (5) map D
mal surface.
isothermically on a mini­
From (8), (16) and (77) it follows that
(78)
J^ [
holds for all circles Cholds.
in
^
,
Therefore (60)
But this is a contradiction of (35), and
therefore the functions (5) do not satisfy (77).
It
follows that the given functions, which satisfy (76),
do not satisfy (77), and therefore the rank of the natrix
JCiu-
fr-u*
UuO“
Kvuu*
tv.a'AT'
~Jla-lMr
X'Vuti”’
(/■(/“
Urn
is less than three. Hence, in particular, we obtain
[
J flu
Kw
Therefore(14) holds.
^ V io T
-
I jf lU L
tvr
From (8), (14) andTheorem A it
follows that the functions (5) map &
_
*
on a surface 3 that lies on a sphere
isothermically
J
of finite
ai
26.
M. Bocher, Introduction to Higher Algebra, p. 47,
27.
If the functions (5) defined a plane surface in
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30non-null radius.
Suppose tliat the functions (5) map D
cally on the spherical surface *5
are mapped on circles.
in
D
is a circle,
be 0
.
such that circles
Then the map of the circle
C* ,on
graphically on the plane
^
isothermi-
S
.
Project
of C
stereo-
and let the map
It follows that P
C
of
has been mapped iso-
thermically on the plane surface
.
By the theorem
of Weierstrass, the mapping functions form a triple of
conjugate harmonic functions:
^
CU.\0-\
J
*■>* *
where (6) and
hold.
Asabove, it follows
(79)
^
that
IU.iA
holds for the arbitrary fixed circle
for tu»o* on
C
in ^
.
But
C ,
which with (79) implies that (78) holds.
that (60) holds.
It follows
But this is a contradiction of (35).
Hence the functions (5) map P
isothermically on
3
such that circles are not mapped on circles,
isothermic representation, then (77) would hold.
have shown that (77) cannot hold;
We
therefore the sphere
is one of finite radius.
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I
-313.
CHARACTERIZATION OF THOSE ISOTHERMIC SPHERI­
CAL MAPS THAT MAP CIRCLES ON CIRCLES AND OF MINIMAL
SURFACES IN ISOTHERMIC REPRESENTATION.
Theorem 3.
the functions (5) have continuous
partial derivatives of the third order in a simply con­
nected domain ]) , then a necessary and sufficient con­
dition that they either map
D
isothermically on & sur­
face that lies on a sphere of finite non-null radius
such that circles are mapped on circles, or be the
coordinate functions of a minimal surface given in
isothermic representation, is that for each circle
in 0
C
,
<8 °>
Necessity.
We have already shown that if the
functions (5) are the coordinate functions of a mini­
mal surface in isothermic representation, then
i.e., (78), holds.
We have also shown that if the
functions (5) map 0
isothermically on a surface S
that lies on a sphere of finite non-null radius such
that circles are mapped on circles, then (80) holds.
Sufficiency.
If (80) holds, then (60) holds.
It
follows that (8) and (73) hold, and therefore, as in
the proof of Theorem 2, we obtain (76).
28.
See p. 29.
29.
See p. 30.
Then, as be-
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I
-32fore, it follows that the functions either map P
isothermically on a surface that lies on a sphere
of finite non-null radius such that circles are
mapped on circles, or map P
minimal surface.
isothermically on a
Here we have used the results of
the second part of the proof of Theorem 2.
But in
the first part of the proof of Theorem 2 we showed
that (60) implies that the spherical surface S
,
of which the coordinate functions are the func­
tions (5), has its first quadratic form in the re­
presentation (29);
circles inP
therefore, by Lemma 1, all
are mapped on circles on S
Corollary 1.
.
If the functions (5) have con­
tinuous partial derivatives of the third order in
a simply connected domain D
, then a necessary
and sufficient condition that
—°
hold for all circles C
point
CUajtte) irj [)
is !)
that for each
,
j={
where
in the circle in D
with center at
and radius Jt .
30.
If the radius of the sphere were either null
or infinite, (5) would be a triple of conjugate
harmonic functions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-33CoroXlary 2.
£f the functions (5) have continu­
ous partial derivatives of the third order in a sim­
ply connected domain
D
, then a necessary and suf­
ficient condition that they either man 0
isothermi­
cally on a surface that lies on a sphere of finite
non-null radius such that circles are mapped on
circles. or map 0
isothermically on a minimal sur­
face. is that for each point ((JUjiTo) in
is, the circle in P
and radius A
4.
with center at
.
CHARACTERIZATION OF ISOTHERMIC PLANE MAPS.
Lemma
curve:
,
x;cu>irt<il|=<rCA*) J
r lL
where C/v
D
Let T
fee a closed rectifiable Jordan
denote the simply connected domain inside T
, and let R
be a closed region that lies in D>
is analytic in Q
closed region
Oi-T
tive number
and continuous in the
, then for any arbitrary posi­
there exists a positive number
which is independent of ^
in ft
j for which
(so)
for all
, i'
“
and
in R
that satisfy the in­
equalities
(si)
it'-tsur.
31.
iv-touJ,
t V t ',51
Clearly, by the uniform continuity of the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-34Proof*.
-V
If £
and t
are any two points of ^
,
_W
, then "by the Cauchy integral formula we obtain
(82)
r
4'-4“
If
4<Qj?
UTc Jr C4-r)v (£-?)
is an arbitrary pre-assigned positive number,
then it follows from (82) that
I
pi
I* e
I - ~ .:p - - * < * " > r T -
.
(83)
if
t84)
|i- 4 " U J , *
where
is the maximum of
length of r
R
and f
and d
.
is the
,
\
on
T
, L
is the
minimum distance between
Here c/> O .
The f u n c t i o n i s uniformly continuous in
hence there exists a positive number
(85)
R
such that in ^
I ^ C 4 10~Vt«ro>) I < ^2.,
when
(86)
| *"-%oU «TX .
If we choose
T -
min(x^i
9
$>. )»
then the inequalities (81) imply (84) and (86).
Hence, if (81) holds, then
(83) and (85)hold.
Prom (83) and (85) we obtain (80).
function Vtt}
in ft , this last restriction can be
removed by defining
^
.
;
t'st!
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Lemma 4.
and the function
If J
) 2.
Let the function
be schlicht.
analyticT in the circle
i a an arbitrary positive number.
there exists a positive number
.
, then
such that the
function
= *5,
(87)
is schlicht in the circle ifcl c
, for all ^
such that
Proof.
If the lemma does not hold, then there
exists a triple of sequences,
(88)
L £/*
(89)
[ 4 m \*.
(90)
I
v;
, >
I " H f * *t> *i'.
with the properties
(91)
/w-*oo
~ 0>
(92)
(93)
^
where
(94)
32.
The analytic function £,(■£> is said to be
schlicht in
if
'<$(*“) implies *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-36From (92) it follows that (88 ) and (89) have
cluster points in the circle
.
Hence there
exist subsequences»
and
1
’
such that
ft
1Iu m
(95)
i
i
ik/wp ~ io ,
^-*QD
(96)
-fiuftA
P'fcoo
I
lLi*p x ,
II
where fce. and £e> are cluster points of (88) and (89)
respectively.
Then
(97)
l4?Uf-S,
and from (91) it follows that
(98)
Hx
m
*|
s°
P -» c e
From (93)-(96) and (98) it follows that
J.Ct'.ir 3, (fc'O
holds.
Since
is schlicht in the circle
l£l
it follows that (97) and (99) imply
(100)
From (93)-(100) and Lemma 3 we obtain
(101)
---
t‘
^ —
—
P-*oo
Since
ttol£
-
rJ<(«L
a o-
^
, (101) implies that the function
is not schlicht in the circle l^«£-
.
From this
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
$t ^
-37contradiction of our hypothesis| it follows that
Lemma 4 must hold.
Theorem 4.
If the functions (5) have continuous
partial derivatives of the third order in a simply
0 9 then a necessary and sufficient
connected ^amain
condition that they map 0
isothermically on ^ plane
surface is that for all closed rectifiable Jordan
curves ^
lying in D
C102)
,
Z [ fft a
Necessity.
’
If the functions (5) map
thermically on a surface ^
0
iso-
that lies on a plane
*1, *v»^I
coincides with the plane X3 a-a
then we make a rigid transformation of the
space such that
and the positive normal at the image of an arbitrary
coincides with the positive X3 -
point C ^ i T ^ of D
axis.
Let tivj.s transformation be given by
(103)
3
Xj - ZL Aic/ Kk
where
+
^
Z Astc AjV
d°4)
*c*i
s-j >
and where the Q.J and
.
S',)
are real constants
There­
fore the functions
33.
C. H. Sisam, Analytic Geometry, p. 267.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
,
-383
X;la.lh) - H
tH
, J ^ I,z, %
(105)
x‘*oi.<rt SO,
map 0
isothermically on a plane surface;
hence, by
the theorem of Weierstrass, they form a triple of
jugate harmonic functions.
From (3), (6 ), (7), (104)
and (105) it follows that the functions
and
con­
x/ (tt>o^
are a couple of conjugate harmonic func­
tions.
Hence by Cauchy's theorem, analogous to (12),
we obtain
t
ia |
®
which, with (105), implies
(106)
-o>
From (9), (103) and (104) we obtain
3
I
(i°7)
I*
“ Z l U | x';cu,vr:)J%\ *
hi
j"->
From (106) and (107) we obtain (102).
Sufficiency.
If (102) holds for all closed recti­
fiable Jordan curves K
lying in P
, then, by
Theorem 3, the functions (5) either are the coordinate
functions of a minimal surface given in isothermic
representation, or they map D
isothermically on a sur­
face that lies on a sphere of finite non-null radius
such that circles are mapped on circles.
Part I.
We first consider the case when the func-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39tions (5) are the coordinate functions of a minimal
surface given in isothermic representation.
he an arbitrary point in
in
0
0
Let
and let the circle
have its center at CU.&,, tTj>).
If ^
Ca
is an ar­
bitrary closed rectifiable Jordan curve lying in D
which contains
terior of 'i
,
in its interior, then the in­
can be mapped conformally on the in­
terior of the circle
CI
such that the image of (.Uo>irV)
in the S, "t -plane
is the center of
C
.
The expansion of the inverse of this mapping function
has the representation
(108)
where
nra.s+t t~ ,
io -U-ott <T0 ,
Since the functions (5) are the isothermic coor­
dinate functions of a minimal surface, it follows,
from the theorem of V/eierstrass, that these functions
may be written in the form
(109)
where
and
X;cu,o-wV[^J^^
<£;(.•&
is a function that is analytic in
its conjugate function.
'.Ve may write
4>JC*> = I I V
^-0
then
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D
-40e» _
,
3^
m-o
where the series for 4 ; ^
and
4^ 1*)
are absolute­
ly convergent in the interior of and on C 0 , and
where 4->,* and
and
are conjugates of
respectively.
(no)
Therefore
- i'*>0 +XL
^ C a,>Ma
*x
/M*l
since
34.
This is a Taylor expansion for
it converges absolutely in and on Co ;
<^C%)
and hence
likewise for <§(&,
35.
Since (108) is absolutely convergent for lunc^ f
M
the series for
is absolutely convergent in
the circle
vergence . p. 75.
, /vt-viji,*** ,
See W. L. Ferrar, Con­
Moreover, the Taylor series for 4;(-*5
is uniformly convergent in £ o , and therefore it is
uniformly convergent in and on
.
Hence the series
qft
TAivkHiVirt-Wf*
•■m
t
all ^
.
(
uniformly convergent, | u H ^
, for
Hence by the Weierstrass double
series theorem (K. Knopp, Funktionentheorie. I, p. 89),
the series ZL
/M-i
the analytic function
Zt>Y* is uniformly convergent to
j=A,V>3.
Since a
similar discussion is valid for the right-hand member
of (110* it follows that (110) must hold.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-41oo
where
C (u.tMA ~ 21 bu;. »•* *b
t<*
*
-
*
•
(1 X 1 )
v
k v+ M » *
** »
M>AW*
If C
r
us
R
of
0^
is the circle concentric with C
, then the function (108) maps the interior
conformally on the interior of a closed rec­
tifiable .Jordan curve £
of &
and of radi­
;
that lies in the interior
then from (9), (108)
Cii2)
f
and(109)
we obtain
* M
rR
c rt
From (108) we obtain
oo
( 113)
-& W ) -
21 ^
Ww u r * * " 1 .
4*3-1
I
After setting UTsRe^ in (108), (110), (112) and (113),
it follows, since the resulting series are uniformly
convergent, that
c
(114)
>2 1
I
®R
must hold.
^
(lis)
g£_
P-1-W4
--1ft £ a;,*
M-l
^
K ,
From (114) it follows that
r r
T -\(
J-t K
oo
op
i
ao
t i.
*l*| S-( J'l
oo
,
LL.
*M=I
holds, where
(116)
$ __ = '“ L
C»,«.
a..***'.*,—
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.
-42Note that
(117)
^
K 1 JL. **
KjA
Therefore it follows from (102), (115) and (117)
that
UlL
(us)
Z Z I I A m > s S M)/v*
i
\
*o»
<•*.* *»*t ^ i | S=.|
where
(119)
AyUjJ** As,(VV “
'
Jv - (
But recall that the circle
was an arbitrary
circle concentric with and interior to the circle
Therefore the relation (118) is independent of R
i.e., (118) holds for all R
* - l l-M*
(1 2 0 )
, o^R
.
C
,
Hence
IH
ZLZLZ1
Aiw>s
v
*
**-» s-( ,*^1
Prom (120) we obtain, by an induction,
(12!)
For, when ■(■*-* in (120) we obtain
(122)
At>|
■
From (111), (116) and (122) we obtain
C123)
A w C t ; = o .
36.
This follows from (111) and (116).
37.
See theorem on p. 170 of K. Knopp, Theorie
und Anwendung der Unendlichen Reihen. 2nd ed..
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-43But the function whose inverse is (108) maps the
interior of & on the interior of C
3?
one manner.
Therefore
(124)
in a one-to-
b(* 0 ,
From (123) and (124) we obtain
(125)
Ai.t - O •
Now suppose that
A/w,s
(126)
where
M
,
and p
are positive integers.
We
shall show that
(127)
holds.
M*3=-p,
For
, (120) yields
P—I
(128)
P - aM
w
2 Z X I ZZa*>s8*>**
**-1 s-/ Avm
~ o,
From (126) and (128),
?-l
(129)
H
A k >v_k ?>*,* & P^ p _ K ^ 0 '
K^-l
r
But (111) and (116) yield
- S k, k,S ,
***'
and hence (129) may be written in the form
38.
A consequence of Darboux’s theorem.
See
W. F. Osgood, Functions of ^ Complex Variable. p. 167.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-44-
(130)
b p Z L KC^
K b
kr=-l
A
^ O .
1
From (124) and (130) we obtain
(131)
teCj?-.obfcL
£
^.p-k'0 -
tc=l
By Lemma 4, for a fixed R
, the func­
tions
(132)
Gr^urt =
+ lv|^ur '
are schlicht in the circle iur\
where
6-i,g
provided
is a positive constant whose existence
was established in Lemma 4.
We further restrict &Jg
to satisfy the condition
for S
£n
p
and
-
^
l
9
where S
, in order that the map
by (132) shall lie inside
is inside
(133)
is the distance between
.
if
.
£*
If of
in
$>R
Moreover, C(U,<To)
Hence, if
ocjiV^l^d^
, Y
r *]»
I-'iV - , P-l,
then we obtain the following result which is analo­
gous to (131) for the functions (132)s
(134)
P-»
(a) L*<.P-*>b|cl- A ^ + i J C K ) U
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
h n
=0,
-45-
(134)
c b ) E « ^
A w ~ °‘
where (a) holds for all
% , /4 S 4 P*<
g-*p/i , and where (b) holds for
p(* .
* except
From (131)
and (134) we obtain
(a)
(135)
to)
S^C1^
where (a) holds for all %
+ a-b £ ^ ^
,
o>
(<* g^-p-l , except
%-tli , and where (b) holds for
Since ^
.
is an arbitrary constant which is sub­
ject only to the restriction (133), it follows from
the relation (135), (b), that
(136)
*|£,? ~ 0 -
must hold, provided p
positive integer,
if
is even.
|c g ( £. p-|
‘
If g, is a fixed
, and
, and
, then it follows from (135), (a), that
39.
The statement has meaning only when p
an even positive integer because I
integer.
is a positive
Hence when we state "except for
we mean that if p
is an even integer, then J
not take on the integral value
&
.
"
does
In a simi­
lar manner, we say that (134), (b), holds for
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A
is
-46(137)
A SlJ(>
If £v
is a fixed positive integer, /a g*4fM
and if
^
*
\>„ a -° , then we consider the function
"'Ok
(138)
H ^.UJrt * <vf
where
,
Vp-g,.
a
ur^u
>
constant,
The function (138) is schlicht in the circle
and maps
on a closed rectifiable Jordan curve
which lies inside
side
iwi - R ,
.
.
Moreover, (Ua,ir6)
is in­
From (108) and (138) we obtain
(139)
where
If we apply (139) as we have applied (108) in the
earlier part of this proof of Theorem 4, then we ob­
tain the following result, which is analogous to the
relation (135), (a):
(m
o
)
40.
aiv(p-gv)4
*l‘ s * C * °>
8v,P-Jv f »-
This holds because IW I c d« ?
•P'S*
’ Ov
was defined in (133).
, where d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
l-iu
-47where
m
Zv
is an arbitrary constant,
(14D
o
i^
gv«
Ox-
Here the constant d|w is in the same relation to
as
is to
.
Since (y|'
and
, we
* iw
obtain the following result from (140) and (141)s
(142)
A f„f-S. = ° -
From (120), (136), (137) and (142) it follows that (127)
holds provided (126) holds.
Since (125) holds, our
induction is now complete.
From (119), (121) and the substitution
we obtain the following relations
3
2— ^
Jit
^
^»»s ) =•O,
(143)
2 -
where the
W)\<w
e ,
■+ °(*» i ^ > **0 =
and ^ **
efficients in the expansion of
about the point CU.6,iTo ) •
00
(144) K;tu.o^ = *:10+ 7~n.*"(
AHil
41.
/V\> S i I j
^
are the Fourier co­
Xycu»o^
, jitjX,'* ,
These expansions are
V
*
%tu ivhs),i -I. ^ ^ ,
Compare with «J. W. Hahn and E. F. Beckenbach,
Triples of Conjugate Harmonic Functions and Minimal Sur-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
—48—
We transform the axes in the
-space such that
the new origin is at the image of Ctlo.tTo') , the plane
is tangent to the surface there, and the positive nor­
mal to the surface there coincides with the positive
axis.
—
This transformation is given by the equations
in (103), where
a;
Ak ;
K-l
and where (104) holds. Then the relations (143) are
invariant uncer this rigid transformation.
For,
since the functions (5) form a triple of conjugate har­
monic functions, it follows that the new coordinate
functions
(145)
3
H
I
A*; Xfetu,*) -
ca*,(r0)L
> * 1> z,i,
i
¥»forma triple of conjugate harmonic functions.
Hence
we may write
faces. Duke Math. Journ., v. 2 (1936), Lemma 1, p. 699,
and footnote on p. 700.
42.
Since the functions (5) are harmonic in P
follows that the functions (145) sire harmonic in 0
From (104) and (145) we obtain
'~
Hence the functions (145) are isothermic.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
»
,
, it
-49(146)
co
x}itt,o*3. H
AMS.O
Since the X'yiUoXi>)
eot»uA
w
am &) ,
oi ^
\J*d
, ,
J > J.
are equal to zero, it follows from
the relation (146) that
v
(147)
«*'. -Oj
*>o
From (144), (145) and (146) we obtain
4~ l .
a
where
'
4-
which, with (104) and (143), imply
C14S)
J
,jV">
^
V
vt
1" (“is ^iw + ^ J,m ?ii)
J
'
and s are positive integers.
/M>S— I
Therefore the
relations (143) are invariant under the rigid trans­
formation defined by (103) and (104).
To prove the functions (5) define a plane surface
it is sufficient that we show that
*'1,0
(149)
holds.
Let
(a)
“ J'vm
Hi
be the positive integer for which
IT
~ o,
(150)
a.)
43.
k < t- 4 0 . ¥ f
J-I
,c
If (149) holds, them.
zo
and- ^
is
a plane surface.
44.
If
for all''4* , then it follows from
the relation in (148), for
, that 2LP*
£
|-V
for all
• *
vvi does not vanish for
SM
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
-50From (146), using formulas analogous to (38), we obtain
(a)
(151)
(b)
- *'„* ?*ct-
(c)
*\,t < t + « C
since the positive normal to 3
- < * ?U - < * $U ,
at the point C0,0,o) coin-
(Vi\-i-66>o , then there is a least positive integer *
for which (150), (b), holds;
45.
Since the functions (145) are harmonic, we may
write )C»ca»o^
tain (146).
3«-
therefore (150) holds.
i
m*o
Therefore
®]
from which we ob-
4
T
. v/henever 21
j-y *U
(MiO
, then
?[.CXfcttXto-“X!cwXlj<|) ^ l c | ^ ) \ A in cyclic order, are the
direction cosines of the normal to S
tions (145) are harmonic, then Z !
.
Since the func-
s°
near CUo^d)
only if these functions are identically constant in
If the functions (145) are identically constant in P
then (149) holds, since the functions have
.
,
expansions (146).
If the functions (145) are not identically constantin 0
then for A,
,
sufficiently small
3
.
Therefore < =[(<* ?t.±~4.t
whereOtn) denotes a quantity (not always the same quan-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1+0(0,
-5 1 -
cides with the positive X 3 -axis.
Equations (151), (a)
and (b), are linear and homogeneous in ^\ct and ^\(t j
since by (150), (b), and (151), (c), their determinant
of coefficients is not zero, it follows that
(152)
o
« \ tt - f a *
From (148), for
(153)
, (150), (b), and (152) we obtain
06'(X *
" &•*
^0,
Cl64)
^tVh* + < *
From (151), (c), (152), (153) and (154) we obtain
( 1 5 5 )
$ ' » *
•
From (148), for s=-fc£/u. , (152) and (155) it follows
hold.
The elimination of
from the relations (156)
yields.
(157)
and the, elimination of $'■»■><* from (156) yields
(158)
+
tity) <€10 such thatl^/w/hl
ly small.
at
t o ,6 ,0)
is bounded for /u
sufficient­
Then (151) follows, since
.
Note that
remains continuous at ^
-I
,
thus insuring the existence of a normal to S
even at
^ 3)C*v
^ 5X<> t
points where 2T(-g-J) = 0
when ZIC
^ 0
. See Hahn and
■»=* ^
/-i
Beckenbach, loc. cit., pp. 701-703.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
-52Prom (153), (157) and (158) it follows that
(159)
From (148),- for
, and (159) we obtain
**
^?jV) ,
from which it follows that
(160)
Prom (148), f
tain (149),
o
r
, (150), (a), and (160) we ob­
This completes the proof for Part I.
Corollary.
If the functions (5) are harmonic in a
fl-fmrnv connected domain D
expansions about the
such that their Fourier
point LUfyiTo)
o£ 0
are given
by (144). and ££ (143) holds« then the functions (5)
are isothermic coordinate functions of a plane surface.
Part II:
We now consider the case when the func­
tions (5) map ^
isothermically on the surface S
which lies on the sphere
, whose radius
^
is
finite and non-null, such that circles are mapped on
circles.
Since the functions (5) have the representa­
tion (28), it follows that the functions (5) may be
continued analytically to map the entire closed d»U'plane isothermically on
on circles.
such that circles are mapped
Let the point P
point £- od , and l e t b e
ponding to
jection
P
as a pole.
on'P ,
thermic map of the Ui<r
on 5
correspond to
the
the equatorial plane corres­
Then the stereographic pro­
with? as pole, induces an iso­
-plane on
such that circles
are mapped on circles and such that the point at infini-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-53ty in the
-plane corresponds to the point at infi­
nity in the plane ^
of
(S
J
.
Here we have taken the center
as the origin in a system of coordinates on
, such that the positive
axis and the ray
Q P
3(.-axis, the positive
-
» in that order, have the same dis­
position as the coordinate axes
(
» Xk*
.
The
mapping function must have one of the following repreWC.
sentations:
(a)
(L61)
or-
(b)
-=>-oCi-f-F ^
6^7 -
C
*
.
U-J
itr,--s.Y-c^y,
where (a) holds if the map of the U)V -plane on ^
is
directly conformal and where (b) holds if the map is in­
versely conformal;
u, and
are constants^
Consider the system of axes
is at Q
XI1,K%) whose origin
, where the positive Xr » xl - and
X* -axes have
the directions of the positive Si -axis, the positive
axis and the ray Q?
transformation of the
respectively.
*i>
origin into the point ^
There exists a rigid
-space which carries the
, such that the positive
X, -,
X-w - and X* -axes coincide with the positive X\K -, xj I
and X) -axes respectively. This transformation has the
representation (103), where (104) holds.
coordinate functions of
5
If the new
are
46.
See L. R. Ford, loc. cit., p. 32, Theorem 25.
47.
When we refer to (161) we shall refer to that
part of (161) that effects the mapping noted in the proof.
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-54(162)
x';~ xi‘tu.>(h\
then it follows from (9), (103) and (104) that (107)
holds, where
is an arbitrary closed rectifiable Jor­
dan curve lying in
D
.
Since the functions (5) map circles on circles, it
follows that the functions (162) map circles on circles.
Hence it follows from Lemma 1 that the functions (162)
have the representation (28); moreover, in (28), the
function ft*}
(163)
x /iu .o -W
is given by (161).
— ^ g|—
Therefore
»
a'-*s,V*«v
xaV.. 1
1 a ^ ^ vl
« j|
From (9), (107) and (163) we obtain
(164)
X
*
where &
is the map of if
on the plane
by (161),
and where
(165)
Since
<i66)
48.
'
%t-—
,
<Pv£s./&)-
—
» fv^ ) - --- ----
, it follows from (102) and (164) that
s*~i
r
Since the functions (5) are not identically
constant in j) , it follows that
ActO.
in (161).
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«
Without any loss of generality we may assume that
*■
£the line w -©
passes through D , where 0
la the
¥■
map
on
curve T
1)
by (161).
Therefore there exist a
with the following properties;
the vertices of T
have the following polar
coordinates^
A* «(A.+u5«>-'c,) ,
2)
T
AV«0ii+u>»,Ti>,
o^r,c1YHj ot-f\t, oztaV,
is composed of two arcs of circles and
two straight-line segments:
circle
ti’" -
, A%A*f
A<AV is an arc of the
is an arc of the circle
MU>i)v , each arc subtending an angle zr,
at the origin, and
A-vA-s and A-«*Ai
are on rays through
the origin*
For the closed rectifiable Jordan curve r
we
obtain
i
-j_
Y~ j j ^(.Si^dur.js /-i
k.(AbU),)~4scui'-& ICM'M) ,
r
where
49. For, consider the rotation defined by
v
(6
UJV=e or, 4 where & has been so chosen that the
line
-o
passes through D
. If <f,-i
avsNC*
and
'K - ----------, then
'
WtC
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
?>»-1
L a.
t
v
q '-+(\,'-11 av4iAi-«i»ov
i
*• OMUrHtiiY'
and where the
1
±_
d''"+A«v
a^i*'
■+rs*- 1
^ (jv*
toi)v 1 ’
a ^ w ^ - -a ^ W
m
^
were obtained from (165).
1*. < • . * . & -
Since
- u ^ . r
L o N iU
i
it follows that
Fix |\-i and u)i so that
v
Since
(^cViu^i
»
.
and
no^ proportional,
6»
y
K . SrtU
^
* ■ * • * >
there exists a ^ <
such that
(IM)
2 1 1)
j =(
r
Now Ct67^ is a contradiction of (166).
functions (5) do not map D
Therefore the
isothermically on a sur­
face that lies on a sphere of finite non-null radius.
This completes the proof of the theorem.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1
-575.
5.1.
MEAN-VALUE SURFACES.
Mean-value surfaces.
If the coordinate
functions
(168)
of a surface S
are continuous in a domain
D
, then
the circular averages
(169)
where
A
^
is a positive constant, will be said to
define a mean-value surface
associated with 5
.
We define
A^lo u u » ~
Xj cu,«rt
,
J -l> *-j i .
7/e note that the functions (169) are defined and
have continuous partial derivatives of the first or­
der in an open set of points Dp
to
D
which is interior
.so
Theorem 5.1.1.
If the functions (168) are con­
tinuous in a simply connected domain D > then a
necessary and sufficient condition that
(170)
J~l
50.
Since
set of points,
c.
is not necessarily a connected
may consist of several pieces.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-58hold for each circle
C
in
£
is that all mean-
associated with 3
value surfaces
32® given in
isothermic representation bv (169).
Proof.
The first partial derivatives of the func­
tions (169) are given by the relations
(171)
£
i
'-rr
> a T' - 'v r
which are valid for points of D ^
,
From (171) we
obtain
s-i
which, with (8) and (170), yields our theorem.
From Theorems 3 and 6.1.1 we obtain the following
result.
Theorem 5.1.2.
If the functions (168) have con­
tinuous partial derivatives of the third order in a
simply connected domain p
, then a necessary and
sufficient condition that they map 0
isothermically
either on a surface that lies on a sphere of finite
non-null radius such that circles are mapped on
circles, or on a minimal surface, is that all meanvalue surfaces
51.
associated with the surface S
See T. Radd, Subharmdnic Functions. p.11.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
„
-59defined bv the functions (168) be given in isothermic
representation bv (169).
5.2.
axes.
Mean-value surfaces and transformations of
In §SA we shall make use of the following ob­
servations.
5.2.1.
is invariant under rigid transforma­
tions in the
-spaces
if
3
Xj -
Xk ^
i
^
*
where the
are constants and where (104) holds,
then
where
5.2.2.
is invariant under each of the reflec­
tions
a-u, u-'z-IT
ana
<r'~v.
If, for example,
then
where
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-60-
(172)
rn. f - J J x'tu'rtjPH,) </{</«(,1 j
J-M.
5.2.3.
Similarly,
^
rigid transformations in the
5.2.4.
is invariant under the
U->
-plane.
Under the transformation
(173)
the mean-value surface
mean-value surface
is transformed into, the
whose coordinate functions are
given by (172), where
Hence the family of mean-value surfaces
ciated with a given surface S'
S^
asso-
is invariant under
the transformation (173).
5.3.
Conformal mean-value surfaces.
Since, by
Theorem 5.1.2, the only surfaces in isothermic represen­
tation for which all associated mean-value surfaces
are given in isothermic representation by (16S) are
spherical maps, in representation whereby circles are
mapped on circles, and minimal surfaces, the ques­
tion arises as to the nature of the mean-value surfaces
in these two cases.
Theorem 5.3.1.
connected domain ^
face 5
If the functions (168) map a simply
isothermically on a minimal sur­
» then each mean-value surface
Sa
associated
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-61vrith
5
is a minimal surface given in isothermic repre­
sentation by (169) and coinciding with
S
for
in t>^ .
Proof.
By the theorem of Y/eierstrass, the func­
tions (168) are harmonic in D
;
consequently, as is
well known, the functions (169) coincide with the func­
tions (168),
A J)fliktA 3
in the open set
.
Hence all mean-value surfaces
associated with a given minimal surface are themselves
minimal surfaces given in isothermic representation
by (169);
moreover,
coincides with S
for tUiuO
in P
Theorem £.2,.2.
nected domain D
I£ the functions (168) a simply con­
isothermically on a surface S
lies on a sphere ^
that
of finite non-null radius (L ,
such that circles are mapped on circles, then each meanvalue surface
of revolution
tation by (169).
associated with S
lies on a surface
and is given in isothermic represen­
Further, for
, pp
is not a
sphere.
proof.
fining
S
hi­
As has been shown before, the functions de­
> (168), may be continued analytically to map
the entire closed Uyir -plane isothermically on the
52.
See the proof of Theorem 4, Part II, p. 52.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-62whole ofjS .
Farther, the functions (168) then have
the representation (28) where
is given by (161).
From 5.2.1-5.2.3 it follows that we may assume
(Xv-
o
>
and
(174)
aoa.
in (28).
point
This is equivalent to assuming that the
P
on
corresponding to the point i-2-00
is the point (
the point Pl
site P
, that
on ^
corresponds to
that is diametrically oppo­
, and that the
£ -point corresponding to
P;CQjb#d) iS reai and positive.
Since we are in­
vestigating all mean-value surfaces associated with
it follows from 5.2.4 that we may take
in (174),
in which case the functions (168), as given by (28),
have the following familiar representation:
t -
xa^ct.
UH
(175)
*1 -
is the circle
+41’
^
t i a (,—
1
V
If <!*
kv
avnrwe<*t ?
u ‘u*o>vs.A'w , and if
U 4 o r = . e td , then (169) and (175) yield
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-63-
C176)
•
Ai^(a»»)+t4a.,
=■e
tdf
i
[4i>pC\oVtAt^pOuo)!.
/»*■
From (176) it follows that the map C v
of
the mean-value surface
sphere
on
associated with the
^ defined "by (175)^ is a circle in a
plane perpendicular to the
ter of
is
*
Ci
0A<
is on the
4*>p -axis.
A-axis
The cen­
and its radius
nf
Since
S
lies on Tp
tions (169) map D p
, it follows that the func­
isothermically on a surface
that lies on a surface of revolution, .namely
In i
5.4, it will appear that
sphere for o c ^ < oo
.
.
is not a
We shall call
a mean-
value sphere.
5.4.
Mean-value spheres.
The mean-value sphere
is a surface of revolution about the
Accordingly, to investigate Tp
-axis.
it is sufficient to
•V.
study the intersection T ^
At,p — O .
of Tp
with the plane
since, by (175),
it follows that
U . A = oj
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-64hence, by (176), the intersection of Tj» with the
plane
can be obtained from (169) by setting
lT«.o
ing
in (175).
A computation yields the follow­
results:
gu * S > ‘>
‘
^
"T" Jo
_
\
f^
*iv+
2<L'-(U+‘flc*5&)nJaj£
“
“ C&S6
““ »
A-
QW
~ ^TL a ^
-f
(
a
^
^
*j #
and
-
rv r
^
- a
T r*°
2^l,u n~<jxd&_____
d
^
/jjf a ^ ^ 1
f
sv
V***
ia»
v
.
from which it follows that the coordinate functions
r>*
of
are
AX
(177)
'
1,
4
- i - 2 £ /„[
y p Q 1-
,
/ _
We make the following observations.
5.4.1.
The curve
Tp
has horizontal tangents
at the points
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4-.,^ ^
At»e-«.
-a*
“65“
A^s.0.-
tej —
-I
,
and vertical tangents at the points
^
i
’T o ^ - ^ l
^
>
a-h£%
A * >e^
v
.
^
i
j
Here we have made use of the following expression for
v .
the slope of
s
1-a CL
^.•4- ------a v-uv+^v
The horizontal tangents occur for il -zo,
vertical tangents occur when
a. -
A-ij(*
and the
.
Moreover,
zT™^'
from which we can obtain the result that the curve Tj>
is symmetric about the line
Am
-
ta
a -
.
In addition, from the representation (177) fori? ,
V
it follows that the curve
about the
Ai,(j -axis.
is also symmetric
From these symmetries, and
from the following relations for the slope
m\
it follows that
5.4.2.
-i
^ o ^ + r
T
q
,
Ma.
u
-
ttkaV^
is convex.
The height ^
of
of
is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
—66—
4
and the width
is
\
v
Therefore, for
circle.
a c ^ < c©
, the curve
is not a
Moreover,
t aco;
x
so that
„
flattens out while approaching the point
Co*4*) , as ^
becomes infinite.
Each member of the family 1V i
5.4.3.
through the point (0, (O
passes
and is tangent there to every
other member of the family.
Further, for [-0
, Tp
is the circle
IU
.
. V
+• A
and for
, Tp
- Q* .>
is the point C°» ^
.
From 5.4.1-5.4*3 it follows that the family
of surfaces of revolution
L T^\
consists of convex
surfaces each of which passes through the point CO,0,<0
in the
-apace and is tangent there to
every other member of the family.
Since the ratio
of width to height becomes infinite with ^
, it
_
*
•«
follows that
Fp
flattens out as ^ becomes infi­
nite.
For
^ —o , Tp
is the sphere with radius &
and center at the origin, and for
point SJLl.Sl
53. For e--i
, T(»
is the
_From_(177) we obtain the following
and the values
is a p(g£ture of the curve Tp
•*
,
there
on the page following
page 68.
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!
-67-
Tp :
representation for
*
5.5.
^
lUl
(v T u t'-uv- T * f )v +H-a H u . ^ ) J
*\
r
Homothetic mean-value surfaces.
If the func­
tions (168) are analytic in the M.CT -plane, a compu­
tation shows that the circular averages (169) have
the representation
00
(178)
Aj U l ^ I I ^
'
* *■' Mil
«
,
■».1,1,4.
t
Two surfaces
^
and
^
, whose coordinate
functions are
Kj'CUjir)
and
respectively, will be said to be homothetic provided
there exists a constant A * o
such that
3y (178) it follows that the problem of determining
analytic surfaces
S
s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
j such
t
-68that all corresponding mean-value surfaces are homo­
thetic to S
is equivalent to determining solutions
of
(179)
21^*
f*i*0
— •
A
Xj
,
In particular, sill solutions
(180)
c.^ x; >
’j*'*1**,
satisfy (179).
For
, we see that sill mean-value surfaces
associated with harmonic surfaces
5
, (surfaces
whose coordinate functions are harmonic), are homo­
thetic to 'S , and, in fact, coincide with 5
,
The functions
/,-i 0L,^IkK IL SOA IT >
(181)
XV
CL
Xv *»- QrtPWL }
which map the
-plane on an infinite sheeted sphere,
also satisfy (179^ with
C ^ 2 -I .
'Ue note that the surface defined "by the functions (181)
is not given in isothermic representation by (181).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
p.o
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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