*
,
is
.
Ai ABSTRACT OF THE THESIS OF
CORNELIUS HENRY 74110
Date.
thesis is presented
for the MASTER of SCIENCE in MATHEMATICS
_.
Apl
8, 1963
,-
Title
ON THE STRUCTURE OF MINIMUM SURFACES AT THE BOUNDARY
Abstract approved
-
1
''
In this paper we consider the behavior of certain surfaces at
certain boundary points. The surfaces under consideration satisfy
a topological definition and are of 2- dimension in 3- dimensional
Euclidean space with the boundary a finite set of straight line
It is shown that the surface of minimum area with a given
segments.
boundary is locally Euclidean at all non-vertex. boundary points.
The key to the proof is a theorem in 1 which itself concerns the
behavior of a set of points under very restricted conditions. It
is shown in 1 that for almost all interior points the conditions
of the lemma are satisfied.
This paper first shows that the part of the given surface
interior to some sphere centered at any non-vertex boundary point
lies near a plane passing through the point in question.
Secondly
it is shown that for any point of the surface lying in the sphere
the surface interior to any smaller neighborhood of that point lies
near a plane.
"Near" here refers to nearness with respect to the
radius of the sphere or neighborhood in consideration.
-
From these conditions we may construct a bounded point set
satisfying the hypothesis of the aforementioned theorem. The
theorem of this paper follows immediately.
a
S
ì1t
i6
^
'
"
!.
41
ON THE STRUCTURE OF
.0
MINIMUM SURFACES AT THE BOUNDARY
by
CORNELIUS HENRY TJORIUR
1,Li
'.r6l,¡r.iA;a5..7
:
,.1, 1
..
o')
;
A THESIS
submitted
0
to
I)' :, :r i5'1
N STATE UNIVERSITY
j
',
;
^'N
('tri
June 1963
,
_
.
.
in partial fulfillment of
the requirements for the
degree of
STIR OF SCIENCE
e
,'íy..R1
"
,
.
1:
f
."
1
.
tr
I'4'
r
.
esd of Qaxint
and
in
Ctin °f Sc1i°
an of
)se
Ct
Cts
$PIIh1!H
áf liaioy.
Graduate ccrsitss
Graduate ScbpOt
si.s 3. 4tes
Zy4ea by S
ea
soä
SI
pr
il 8'
1963
's
AC1CNOWLSDG!021T
',
author wishes to acknowledge the aid and understanding of
Bristol, without whose help
The
Mr. S. R. Reifenberg, University of
this paper would not have been possible,
.-r;NV
.t,.
.
..
;
;.
:.1;..
{
e
4
ON THE STRUCTURE OF
MINIMUM SURFACES AT THE BOUNDARY
The problem of Plateau was solved for surfaces of varying type
One of the problems considered previously was to prove that
in [1].
for almost all interior points the surface of minimum area is locally
Euclidean.
In this paper we consider surfaces satisfying the same
definition as previously but with added restrictions.
For a detailed
description and some examples of surfaces in this class the reader is
referred to the original paper.
Therefore such a discussion will be
omitted here and only the definition will be restated.
Definition:
Let
G be á
compact Abelian group,.
set in n- dimensional Euclidean space and A
g
be it
group
jdrr-
Let
j/ jut
class
j. A
,ice
closed
closed subset of S.
Let
x
.
3 Lt written miss)
.
Let x be the
g the inclusion blialanhin
I
l
;
S be a
Then there is defined the, Cech homology
non- negative integer.
[s A. G)
a
Let
.
m- 1(A ;G) --*1Ç
any subgroup of
4
EEO)
AG
daD L if D
with boundary
Then we
.
se
S is a surface of
L.
We will consider here only surfaces such that G is the group of
integers modulo -2, n-3, and m-2.
Further we restrict the boundary to
be a finite set of straight line segments.
We will show that the sur-
face of minimum area with a given boundary is locally Euclidean at all
non -vertex boundary points.
Notation:
I.
Throughout this paper we will adopt the
following symbols:
S(x,r) is the closed solid sphere, center x, radius r.
.
J
2
x,r)
is the surface of the sphere.
o is a minimal surface satisfying our restrictions.
1(x,r) @ So(x,r).
(X,r) is the set of points whose distance from X does not
exceed r.
$0414,$).
Z(x,r)
B is the set of boundary points of So.
V is the set of vertex points of So.
-
0
AtX
dimensional Hausdorff spherical measure de-
is the
Suppose X is a set in N- dimensional Euclidean space.
fined as:
ri<
Consider a set of spheres S(xi,ri) such that
X
Ci
S(xi,ri).
Let
/\dX be the lower bound of
taken over all such sets of spheres; where
and W
m
- IT
if
m
= 2.
Let
A mZ _ lim
¡-ro
=
W
m
P71
b
and
j'
.
2
E
w
if
m =
1
X.
S(P0,R0) or S(Po,ro) will have no vertex points in the
.
interior.
Given
Lemma 1.
ro
r
°°
E
-rr
)
r 2
__.
y
E
1
where Po
>
0 and S
E BAY,
-
exists
there
an
r> 0,
-,o--
such that
< n 2R
2
+
E
.
The proof of this is similar to the relevant parts of
:
Chapter 5.
Lemma a.
hEif B ) where
aEs(x.r)
[1]J
twit
,k,I
Skg
á surface contained ia
B consists of m rays
bE
(x.r)), plus a set
R1,
= xa
L(x.r) C
S(x.r) with
á,s,d
Ar =
-
s(x.r).
1.
boundary
xb (where
If
fi
az
3
a
4 ira
Aux-0 <
0
)
VII...
ü)
S*C
Cj'_
follows treat
eiromrif
,.
A
it 3
141)(r.»
.,ta tt. Y.Act Ofetti-,. à
This
thus
tith0
.
Ots
+
ii6(r.r)
* caftI +
141)
purely topological considerations.
Y-
Apkt
of the
The alternative
`` w..i
Shea
?,
!s .
P,
s,
..
t
Giran sox E > o
<
E
r - -E md , nL4,,(:.r) < Tr xE
Trr<n
_ 3
.
IS swat labial e tra k
s* ink Sita sat bail= sash that
n i(a.r) > -r-r.
r,
Corollary A.
_
dmoklb&
.-_
above
s.v_
lama being excluded by lama 1.
tJllEt
Wait
r.at ask
E.-
bTrrb2(1 + E)
[7\1
0
.
.
.
bre <ÉÇ re,
Trt < nMAIL--- 41T;
n 1(:o;t} <.
Write
IT
t+ MO" r,
re
°:
f AMAX) t
t
ro
<_
C
and by
<
o
(nt + ,1Udt
3/41.0
r:
Then
Z+ (adt
'o
lisimeowds
fre
adt
o
Meet
<
r2 E #
rrtE for all
.
t.
fre
it dt >
o
contradicting (1).
3/471-re2E
than
(1)
4
Po
Lemma A.,
2
ha
then
11-11,
c
And
x(Po,ro)
Eig.Q,
rrro2(1 ±- E)
C re
.
4rrrz < [AtPGPt# < Trr2.
+
E4).
0
Let
fAl(P0,t)dt . F(r).
ay [1, lemma
*]
o
r 11(P,=)
F(r)
°
iir
t' (Y)
hence
1=(r)
F(r)
and
>ir
integrating from r2 to r1 where
F(r 1)
log
r2 < rl
ro
we
obtain
.r12
> log
r22
(r2)
and hence
i)
.
02=
r12
and
r22
Q
so F(r) /r2 is a monotone not -decreasing function of r.
toms
4]
jAI(P0,t)dt s
gm
for
- <
A
(Poor)
1/21-r(1 +
E4)
r2
r2
r2
Et,
!.
and the hypothesis
il(r).
sv
all r <s*.
jams it.
P.,aE
=r,
o
E
11
kTrra2 < A2K(Pv.ro)
S 1/27 r`,o21 + Eg2)
-,
then
.gps
11.10
ti
5
<
41112
soma 4
By
for all
r <ro.
that
3/4 1-r
<ro
r2 < nR(
there exists an
Then
n211(Po,rl)
hence for
all
t
< t
1+
>
:
so
+ E
(2)
< ljrrr2(1 + 4 E5) is false for
.rt < ro such that
5)
4
4 E
. r2
/4
E5
n2 (P..r
rQ.
.
f
f 4E)
> rrrl
3/4
>
.
,
3/4T-rt2(1
+
3/4
E5)
that for all t satisfying (3)
n1(P
> T7t(1
+ k
Es)
and hence
r rl
r,
fAl(P0,t)dt
o
.
,
ffAflP0t)dt +
«
g2) >
+
If
exists
if
3/4r-r'r
E
<3
E 1R-P
and
which contradicts (3)
then. there
/
rz
Al
+
-
tTr(1 + E5)(r22 - r12)
o
k1rr22(i
6.
/
r,
.
> [/1(P0,t)dt
and using (2)
Le ®a
.
o
"
`
'IT
r
- r12)
our assumption is false.
+ k -rr(1 + k E5) (r22
and hence
some kalif.-plans
ü(S(Po t))_ C
i
some
"
that
<rl ( i+
A2x(i
rr2(1
k
1/2rrr12(1 + 4
such
4 E 5) .
.
fAi(P0,t)dt <
} r2
Assume
21(q.e.4- IL.7r2(1 +
_
A
_
<- .Trt(1 + E62)
r*(PQ.t through, ro, such that
(3)
.
t
6
Awl
C (f *(Pc,
1
Let
n
) . E6t).
be the plane through Po orthogonal to BS(P0,t)
BS(P0,t)
r(P0,t) be any plane through
D1 *(Po,t) r(P0,t)
f
Ì' *(P0,t)
o.
.
BS(P0,t)
extended and
be that half -plane such that
r *(P0,t)
' 1*(Po,t)
O.
By use of elementary spherical trigonometry it can be shown that for
.
any point x
E 1*(P0,t),
then
xE(r *(P0,t),
E6t).
r
Henceforth, the symbols
.
(Po,t) and
r *(Po,t) will denote
this
plane and half -plane, respectively, as constructed in
t¡°'r..y
.
a
a
the
q<'
t*
E
,T
.
'..
-
class
t*
ca
S-V,,
:5-/
Pß
o
s. rrr 2
rM
< _T-rtsi.+
<
.(
I,
-
l U62
<
E
,numbers t=r
Al(Po.t)
Let
r
E
If
2
T
a
fauna.
.
Lemma Z.
.
let
such that
(P0,t) is divided into two half -planes by
let
and
+
_
and
E7.
such that
(27b.
G7)
E)
2
MILL
..
tAen
l*(PA t) C (r *(P,, t )125 yt*)
S
(4)
faiT_
less
shan
.
(Po,t). _ *(Po t) a(P t) and let C be any finite set
components of (Po,t) + 1(P0,t). We can find an open set Gt such
.; :i ,;,
,,,tt :,.. ..>y C C Gt ( -1((P0 ) + 1(Po,t)) CO, E 72t)
,ana'
nG. ( (Pp,t + l(Po,t)) ¿ AC E7 2t.
and
Let
('
.
y,
of
that
;
,
By an
application
.
of the Heine Sorel Theorem we can find two closed
disjoint sets !* and F
.
.,".
,
such that
.
.
!.
J.
t.
0, r'
t ;
-
7.
s
.S.
il.
7
C
C
We +
+ 1(a
(5)
+-S(P0,t) E72t).
+ 7S-(P0,t)Ch#*C(1*NNN,t)
1 *00,t)
and
#C t. ( .6(P0,t)
1100
By (4) and lemma 6 we see that
E72t
lies within
of
-4f
(6)
(Po, t).
The elementary domain consisting of the points of s(P0,t) lying within
2
E72t
of -S(Po,t) will contain F* and moreover this domain is con-
tractible to a point so that by lemma 2A
there will exist a
[1]
surface J *(Po,t) with boundary containing p*.
A2J *00, t)
lama
By
<
Hence
21T E 72t2,
(7)
2 and (4)
nl*(PQ, t)
Tit
E
and hence by (5) and (6)
/17
*<
TT t
E74.
By E1, lemma 8] there will
convex hull of F
exist
with boundary
A2J *(P0,t)
<
4 ,/3
J H,(F
(n f*) 2
5:1
)
Ó(Post)
and
S(Po,t *)
planes
-
K(P0,t) +
-
S(P0,t).
('(Po,t*)
and
t*)
00,t) plus
Let
(I)
the
be
Then by virtue of
is: a surface
[1,
with boundary
*(P,., t*)
-
+ J(P0,t)
J
is
a
surface with
,
angle of intersection of the
.
K(Po,t) + J(Po,t*)
11-702
-
s
+
IInt2
(mediate by projecting this surface
ing lemma 5 to the hypothesis we obtain
4.
(8)
two intervals of I lying in
Then
rl*a,t).
.
is
r
r,
?(P0,t*) +
/\_
This
such that
in the
+ 84(P0, t) .
Now K(Po,t *)
boundary
K(P0,t)
11A], J(P0,t) +
lying
err 2t2 (E78.
4
Let J *(P0,t) + J*(io,t) - J(P0,t).
lemmas 13A
J *(Po,t)
0
a surface
'
,
o,t)
cos 0,
onto
r(Po,t *).
:.. (9)
:.
WY.
a;
8
<Tr t2(ï
A1/410, t)
+ 4E
2'
for all t< r0. Together with (7), (8) and (9) and after
manipulation
obtain
we
sin
(t)
< 25 E 7
t <t*
as long as
and
t >E7 t*.
follows from elementary
if
While
t
Thus sin
geometry.
some
E7ht* the result
<
(N
25
E
7
for all
k
8<t*'
lama .8. 11
CB
>
A2x(P
112-1102
r <ra,L4
h
=.
7
and
(+
,
2±
CrY E s4 and by lemma 4
Tfte2 <
AG
be
E
the set of
< hilt
for all to < ro
/1l(Po, t) dt
3/411-t02 +
all r_ to< ro/2.
Then
to2,
there
(
exists
such that for any rE(=-fg)
fr r < /11{tw, r) < grit +
4
(10)
Eta
wo
Let
).
E
.
juz,#rLÇ*ito.a
Let
.
< Tr
o-
_then .fax
<E
E8
Por, if not,
a
set
GCE,
114
2113/4r.
(12)
r'
then
4.
nl(
t) at .
which contradicts
'
,l
>
A
(11). Hence
nl(P°,t)dt
..
$
4
4,t)dt
+
IAAl(Po,t)dt
- AG) + nGOirrto +
3/4Trt
l0
'
,
Al(Po,t)dt -
t
f
Oto
Al(Po,t)dt.
9
,
Therefore
nl(P®,t)at <
%Tr% + htoA
.
<
to2 + ht
3/4n
to2(h +
37 the
corollary of
and
lemma 2
f
Et4
+ Tf
Trtdt
ho
Trtdt
-
Tf h3/4).
(13)
4 and 3
lemmas
JAL*(P.,*)4s *
8G
J
nl(py,t)àt
f Al(Po,t)dt
-
c
.
? 02(3/4-rr
Let
tEt340
and
assume there
igtriottr
by leers
7, for
all
know
2h'
).
such
(14)
that
r<S*, and rEts.4)
r) ,.
(13)
0.
that
n2[(A,
er
"TT 1141
ur,
a point
nl*ts .t)dt
(
0) , 26 E t0)..
ti(X. 25
We
is
h -
J
1R-c
.
by
2
E .t0)
?
=2tp2-rr .
lemma 3 and (15)
;Trto
+
E
. 210E
I( . 3(Esto)
(so'to)
A
.
>it.a AMP
,r)
> tostcTr - h
which gives a
'
ItrQ,
This
contractictioo
concludes
r0/4)
lies
if E< 1.
that whenever
in a narrow
Lamas 1
TT
210I)
h3/4
J.
holds for
Stra,
than
slab centered about a constructed
plane
10
through
We
It is will
TO.,
will
row muddles below
contained
.._.4t,
Auk%
.
2
be
(I)
l
the mapping
111
km IL
imi
is
2
a
reducing.
D
E
v-Tr2 <
i;
x
[stro.ro) - s(rl.rl)
,
j-
0
.
?-1S
± E10) iki8
1.2.40)
tit Z
1, -_
r
'Pit
9 with
whg230.3
isral
°C+s
O.
mmtlit
Iv,/ :LIAO
tturall t Eu g
nllcri.ar)
Tt!
The !!voi` oasis
.
(
Place every terms
< Ala
4 TT
-
...
_
follows
Tt
_
t fi
_
111** incidentally that
Pi
baling*
of I.
Isms 11. Vasa E",Q Shia
El);,
.
lama 9 with
from
to the
as the
origin
surface
and
ease it
.
I
.
(
Ail
1/212+312,z'0.
direct application of laws
s1Ase. L
is a
igres) -
D
,n?itc
11.
looms ...
r.
area
14t1,rl)i
harif
proof
I
flowis
ihe)
The
r
mete' 8'
[We
lift,E
AH
(yl,e
T'
reducing and thereat&
which links
c
.7Gs
,_
+ Lifer.) ._
k
to 616011011
LAM .40*1
,,..ii
-dgiastts,
w..i..riCr
111.040) S TlY
hikes= ,n UMW*
,. _
*113
distant*
the surface interior
of
:f
far
boa thus
in stro,te).
MONA
Let
!o
note that
ì
-.1.2->.-112. suf . :. re!°::'C,,
.
11
Mg
far
every S(P',r)
C S(Po.ro), P'Ç10,
< n K(P' ,r) <
Tfx2
T1"
2(1 +
á plane f (P$,
then L21 each r<1(16 ro there ,exists
4
E2-)'
g
CI r'(yo.r) . E *r)
f(Po ,,r)
a.r) E (X4PQ. r) .
K(P8,r)
is
This
.-...;
E *r) .
proved in [1] towards the end of Chapter 3.
,.
..
12.
Lemma
such that
P--
4Tf(16Ro)2
EBv'
<
E13< Rua
A21G(P,16R0)
E5-'
E
1Ef
¡'
k--8
and
E135
IrTI'S1610)2(1 +
then far each R<R and each PEK(Po.
©) there exists á plane
through the segment
the boundary through such that
.:,.
f
(P,R)
,f
Ç(¡(Px.
.
RICA
The proof
Case
is
"R,
;
Ei,R
S
P it .
by using cases of P and R:
;
I.
PEBY;
.
R <Ro.
By lemma 10
II
tr ( 81to) 2 < A211:0,00 <
1-111010) 2(
+ 4
.
E.
Thus by lemma 5
'
3/4fR2
<
n2K(P,R)
and hence by
II. rE(So
+ 16 E134)
(16)
8
C
K(P,2R)
Case
< 1ïR2(1
( *(P,2R),
(17)
28 E13R).
- B)1100,110) and
IP P1
each Pj corresponding to each PE14P0,R0)
<
R <Ro,
By
(17) for
O
.
K(P , 2R)
C (f (P :2&)
>
Let ('(P,R) be the plane through
K(P,R)
Case IZI.
C
PE(S©-
( r'(P,R), 29
28
P
E11»
and
repro),
parallel to
Ella).
B) s(Po,R0) and
LP
then
(18)
PI
> R >1/16
I
P
PI.
a.
"
»,
r
V
From (18)
.
12
.
4
,
Irirl ) C
R(P,
P,pI), !E13 )
(
then
1C(P,R)
P jP
I
irr. PE(so
r
I
IP"P'I 2
E
.
all
) for
. 011(l0a0
lP
I
R
in consideration.
R<1/16
and
}
P'E TAP,
corresponding to every
2
213
(
r(P,R)
where
cage
C r(P.),
I
By
IPPI.
2Tr IPi1101I (; + 24
<
<_
j
r
E13
.
t
by lemma 11
rrIPP'I
for
)
7
then
(16)
2
<
A
'
< rr
I
P'I (1 + 25 E
13
4;
and by lemma 5
-
TIR2
AZx(P',R)
Apply lemma 12 with
-
r
-
flit
R and
(P, x)
(P,U C (r(P,R),
sl
G
(x(P,R),
E13* so small that
*<
r(P,R)S(P,a)
,
Taking
m
+ 25 E132).
to obtain
Eft)
,
and
ro
(1
Eft).
min
,
(2'13
(1:*)
the desired
result is obtained.
".,
.
We now
,...
quote
a theorem from [1] with the
.
notation established
I
in this paper.
Theorem:
-
point of G such that to each
corresponds a
plane r(X,R)
,
GS(x,R)
and
t
g
and
f
r
set of points
If G is a bounded
in B3 and Po is a
-
C
itt<ii
o
th
and each =EGS(P0,80) there
0
X
such that
(r(x,R),ER)s(X,R)
S(X,R)
(X,R) s(x,R)
C
(G,
ER) S(X,R)
is a plane through Po such that
'
(A)
(s)
,
-
13
(c,Ex0)».
Then if E
<
(C)
there will exist
2"1
a
topological
2 -disk
G such
that
/16 gp)
N
C IIC GS(Po,Ro
We will now construct from
a set G and then we will
:
show that G is a set which satisfies (A), (B) and (C) above.
a cylindrical coordinate system P(r,0,a) with
P(P0,10) be the plane
z
w 0 and
1400, 10)
o
Define
,
w P(0,0,0).
Let
be the line 9 w 0
sr. e.
Let F be a function whose domain is K(P0,R0) and such that
F(P) r (r,w9,.e) for all PEX(P0,R,).
Let G be the set of image
points of F together with K(P0,R0).
First we need another lemma.
Lem pa
j.
Suppose
xEX(
Stu
/2)
ham
,.t
Xt
Filltcruto
by virtue of lemma 13,
through B.
it and
f2.
B will divide f
Cella)
t
R(X,
into two halves
both in
;Road.
r which
f1S(X
has X near
19/10 hail) or
is the projection of
some
For elementary topological reasons we could else
point of 1(P0,ro) in F(S(X,R)), if such exist.
f.
.
In particular there
form a closed torus linking B but not meeting K(Po,ro).
'
Then
IXXI) will lie near a plane
('2S(X), 19/10 IXXI)
point of K(Po,ro).
161,.
-
s.
Now either every point of
every point of
ç
I
woad.
F(S(X,R)) will cOntain no points,
will bn na pair of distinct
.4
i
Let Y be a
both
14
/ 2 1's(x
f
and
n2
,
19/10 I1' 1) +
A (TI
+
/2x(,
r2s(xj, 19/10
will substantially exceed
I
%1-rI
Xxjl
,%I
I
xxja
I
zcI2`4 and one of them gives a con-
tradiction, depending on which is fully covered by the projection of
K(P0,ro).
Now suppose that XEK(Po,ro) and that s(X,R)
aider first the case where 1xjI
<
C S(Po,ro).
Then by cases I -III of
1/16 R.
lemma 13, x(X,*) will lie within 213 E13R of
f
(Xj,
this plane passes through B the same will be true of
14
Thus all points of GS(X,R) lie within 214
X parallel to
(l(Xj,
I
Xxjl
Con*
Ewa
ixZjI).
Since
mi., -1(7.8)%'
of the plane through
).
On the other hand consider the case where
IxXji
>
1 /16R.
By
lemma 14 and lemma 13, case IV, our condition is then satisfied.
Thus by the theorem quoted, G is a disc near Po.
'
Now again quoting
lemma 14, this disc is divided into two closed congruent sets whose
common part is B, namely =00,ro) and F(X(Pe,ro)), each of which
must therefore be half discs.
This completes the proof of the following theorem:
Theorem:
For all non
boundary points of So as defined
0
in the introduction of this paper, the surface is locally Euclidean.
BIBLIOGRAPHY
1.
Reifenberg, R. R. Solution of the Plateau problem.
Mathematica 104:1 -92. 1960.
..
. .
'
,
of
.l
..171
i'
Acta
"
.nt
;
g
'
,
w
a
`