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Some strange properties of minimal surfaces in connection with Plateau's problem
Computer methods in applied mechanics and engineering Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
ELSEVIER
Some strange properties of minimal surfaces in connection with Plateau’s problem Viorel Institute of Solid Mechanics,
Visarion*
, Margarit Baubec
Str. Constantin Mille, 1.5, 70701, Sect. 1, Boukarest,
Romania
Received 15 June 1995
Abstract In this paper we shall prove the existence of some minimal surfaces passing through enough smooth curves, surfaces which satisfy some given boundary conditions. We shall point out a strange property of ‘cutting’ of these surfaces, at least for large classes of curves and boundary conditions. The theoretical results will be completed, on the one hand, by a graphical study, to construct and to see the surfaces, and on the other hand, by a numerical study, to obtain the geometry of the cuts.
1. Introduction From geometrical point of view, a minimal surface is defined by the fact that in every point the mean curvature is null. In the sense of [l], Plateau’s problem consists in finding a portion of continuous minimal surface without interior singular points, terminated to a given and closed curve. The minimal surface 1ocuZZyminimizes (or maximizes) the area, among all the surfaces, passing through a local close contour lying on the minimal surface, which can be generated in a certain spatial vicinity of the minimal surface. Further on, we shall develop a method to generate continuous minimal surfaces passing through a given, fixed curve and we shall demonstrate accurately the existence of the solution, at least for large classes of curves and boundary conditions. In the case of a closed curve, if these minimal surfaces have not interior singular points, they are solutions of Plateau’s problem. For a class of curves still large enough, in which the parametric coordinates of the curve are given by trigonometrical polynomials, we shall bring into evidence an exotic property of ‘cutting’ of some surfaces, after a continuous zone lying in the vicinity of the curve. After this zone, the surface is ‘broken’ in a number of ‘petals’ evolving whimsically in the space. We completed this special study, in an accurately mathematical point of view, with the achievement of a package of programs, for graphics and numerical calculations, necessary on the one hand, to construct, to see and to rotate the studied surfaces and on the other hand, to solve on a numerical way the equations of ‘cuts’, which, as we shall see, cannot be solved directly except for the simplest cases. These programs were written in FORTRAN, calling however subroutines and functions of C language (especially for graphics).
* Corresponding
author.
0045-7825/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved PIZ SOO45-7825(96)01118-S
312
V. Visarion, M. Baubec
I Comput. Methods Appl. Mech. Engrg.
141 (1997) 311-333
2. Construction of minimal surfaces To simplify, we make the next convention. In few places in this paper, we say that a vectorial function satisfies a property (or a relation) which was defined for real functions. This will mean that the components of the given vectorial function satisfy the respective property. Let 0 c [w C Q= be an open interval and c : O+ R3, c(O) = (c,(O), c,(O), c3(0)), 0 E 0 a function which gives the parametric coordinates of a smooth enough curve (C) in R3, admitting in every point c(O), the existence and the oneness of F&et’s system of vectors. So naturally, the following conditions must be satisfied: (a) 4 )ECmQ) (c h as d erivatives of any order in 0). (b) The vectors {C(O), i;(O)} are independent linearly, for any 0 E 0, where we note C=a
dc
i;=- d2c de2
(c) 3 IY> 0 so that (YG IC(O)l and (Y< I(C x i)(O)1 V 0 E 0. This condition is not artificial, because one verifies immediately that (a) + (b) involve (c) in one of the following cases: - the function c( ) is periodical, with the period less than the length of the interval 0. - we restrict our theory at a compact interval J = [c, d] C 0. When (C) is a closed curve, these two cases take place in a natural manner. Let t, II, b : O+ R3, t = t(O), n = n(O), b = b(O) be Frknet’s system of vectors in a point c(O). It takes place: t=;
i x(i
XC)
n=IcIIcxtl
(i’xi’)
(2-l)
b= Ic xi;1
where lel = j/e: + e; + e: is the Euclidean norm of an arbitrary vector e. We try to find a minimal surface (S), defined in a domain D C_R2, conveniently chosen, given by x : 6+ lR3, x =x(0, v), in isometric coordinates, so that the following conditions should take place: (1) The curve iies on the surface, thus x(0,0) = c(O), V 0 E 0. (2) The tangent plane to the surface, in the points of the curve, must be established by the tangent t to the curve (or by C) and by a perpendicular vector to c, noted a, which must have the expression: a(@) = lti(@)l(n(@) cos P(O) + b(O) sin P(0))
V0 E 0
(2.2)
where the function P : O+ [w is at least continuous. It is obvious that a(@)lc(@) and la(@)1 = [C(O)/, V 0 E 0. Further on, we shall prove the following theorem: THEOREM
1. Let a, c, P, t, n, b be the real functions given before and f a complex function f : 0-+ C3:
f(0)~f~(@)-iu(O),
VOEO
where
i=fl
(2.3)
We assume that there is a simply connected domain ID and an analytic function, h : IID+-C3 so that UCD&_@ and hl,=f. Let DC[W2 be the domain: b={(0,v)EIW2/(0+iv)EDC@}. Then, there is an analytic function u : [ID+ C3, ri = h so that x = x(0, V) dzfre(u(8 + iv)), (0 + iv) E D, defines in isometric coordinates a minimal surface (S), furfilling (1) and (2) conditions. PROOF. Let us take u : ID-* C3 a primitive in D of h (established, x : !D+- lR3 x(0, v) = re(u(O + iv)).
excepting a complex constant)
and
V. Visarion, M. Baubec
We have, obviously, du -=p dt
I Comput. Methods Appl. Mech. Engrg.
using Cauchy-Riemann
relations:
d.x d.x d re(u) + i dim(u) ---jg--=~-i~=x,O d@
d re(u) + i dim(u) dt -= dt
313
141 (1997) 311-333
-ix,,
(2.4)
We know that for the surface (S), the vectors x,@ and x,, define the tangent plane at the surface in every regular point x(0, v). On the other hand, we have du = h(O) =f(O) dt ,,=,, OEO
= C(0) - iu(0)
(2.5)
From (2.4) and (2.5), it results in (2.6a,b) The relations (2.6a) and (2.6b) mean, in fact, that condition (2) is fulfilled. Also, integrating one obtains x(0,0)
(2.6a),
= c(0) + ct.
We remember now that ~(0 + iv) is established, excepting a complex vectorial constant, so x(0, V) is found up to a real vectorial constant. Conveniently fixing this constant, we can satisfy condition (1). ‘Now, let the analytic function g : III-C be defined thus: g(t) = h?(t) + h;(t) + h:(t)
(2.7)
We can write S(%EO =ff(@) =6:(o)
+fi(@)
+f:(@)
= (iI
- ia,(@
+ (&(O) - ia,(@
+ (C,(O) - ia,(0))2
+ c$(O) +6:(o)
- (a~(@) + a:(@) + a:(@)) - 2i(k,(@)u,(@)
+ k,(@)u,(@) + t,(O)u,(O))
= IC/*- la/* - 2i(d, u) = 0
(2.8)
where (C, u) is the scalar product of the vectors C, a. Due to the oneness of the analytic extension of a complex function in a non-empty &LEO = 0,
it results that
g(t)ltED
=
domain, from
0.
One obtains g(t)=(s)*+
(%)*+
(s)*=O
in I[D
(2.9)
From the last relation, in accordance with Weierstrass’s theorem of representation for analytic functions, it results that x(0, V) defines in b a minimal surface in isometric coordinates and finally, the theorem is completely proved.
3. The existence of minimal surfaces In this section we shall give sufficient conditions for the existence of analytic extension of a function, so that it should be possible to find some classes of functions c : I+- R3 and P :04 R for which the existence of minimal surfaces can be rigorously proved. We rely on the next result:
314
V. Visarion, M. Baubec
THEOREM
I Comput. Methods Appl. Mech. Engrg.
141 (1997) 311-333
2. Let the function p : O+ R, 0c R, p E C”(U) be such that
3a>O
so that
YiPi n
limsuo-
1
n
(3.1)
Ip’“‘(O)l =sup
QEO
Then, there are R >O and an analytic function (-R, R)}, so that gl, =p. Moreover, g is given by the next series: g(0
g : [E-C
where
[E= ((0
+ iv) E @/O E II, v E
+ iv) = g,(O, v) + ig,(O, v)
where for any (0 + iv) E IE:
g,(O, v) =p(O) g2(@,
v) = +
-;
p’2’(o)
p”‘(@) _
+$
p’4’(o)
- ***
$ p’3’(@) + $ p’y@)
(3.2a) - ...
(3.2b)
PROOF. We shall show that the last two series (formally for the time being) converge uniformly in any compact subset of a domain E conveniently chosen. Weierstrass’s criterion of uniform convergence shall be applied. One takes the following rectangular domain: vE(-R,R)}
E={(O+iv)E@/OEO,
(3.3)
where R=-
1
ez2.71..
a2e2
.
(3.4)
therefore, we take E = ((0, v) E R2/(0 + iv) E E}. Let us take KC E a compact subset, then there is V > 0 so that Kc ((0, v) E R2/0 E 0, VE[-ti,C]}
and
i
Because V < R, there is p > (Y so that 1 i=-----p’e’ From the relation
(3.1), it results that for E = (p - cr)/2 > 0 and for p = (I! + E, 3 n, E N* so that
Ip’“‘(O)l < pnnn, V 0 E 0, V 122 n,.
In these conditions, it is obvious that, except for a finite number of terms which have no importance, the series (3.2a) is absolutely minorate, for 0 E 0 and v E (-Z?, +R), by the series: fi+(2p)2$+(4p)4$+(6p)6$+
where Z? is the convergence R=
1 li,m supm
...
(3.5)
radius of the last series.
where
b, = (2np)‘2”’ -$q
so
1
R=-_>v p2e2
It results that (3.5) absolutely majorizes (3.2a) for 0 E 0 and v E [-6, C]. Now, all the more, as it is clear that the numerical series:
V. Visarion, M. Baubec I Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
315
-2
p+(*p)2g+(4p)4$+(6p)6g+*..
(34
is convergent, while the series (3.5) and also (3.2a) excepting a finite number of terms, are absolutely minorate by (3.6) in K. In accordance with Weierstrass’s criterion, (3.2a) converges uniformly in K and further on in any compact subset included in 5. Similarly, we have uniform convergence on compact subsets in E for (3.2b). Moreover, we can see that the series obtained through partial derivations however more, are of the same type with the initial series and they also converge uniformly on compact subsets in E. In accordance with a well-known theorem, it results that g, and g, are continuously differentiable any number of times in E and their derivatives are given by the partial derivatives of the respective order of the series (3.2a) and (3.2b) (which give us g, and gz). Further on, one sees that $
=
%I -=__-
p(e)
_
1”!
g
p’3’(@)
p(@)
+
+
g
$
p’(@)
-
#y@)
-
. . . =
. . . =
-
2
(3.7)
$
(3.8)
av
As we know, whether in a domain (E in our case) g, and g, have continuous partial derivatives of first order and they satisfy the relations (3.7) and (3.8) in E (named Cauchy-Riemann relations), then the function g : [E+ C, g(0 + iv) = g,(O, V) + ig,(O, V) is analytic in IE. Finally, one observes that and
g,(@, 0) = P(@)
g,(O,O) =0
VO E 0
(3.9)
so gl, =p. The domain IEand the function g being established, the demonstration is complete now. Essentially, we have proved the next result: ‘Any real function, defined on an interval of the real axis, having derivatives of any order, and that has successive derivatives that do not increase too fast, in the sense of relation (3.1), admits an analytic extension in the complex plane, at least in a symmetrical vicinity of the real interval.’ Also, we can prove this result, observing that such a function ‘p’ is analytic on 0 and using its Taylor series. A special interest presents a function p : O- R, for which (Y= 0, so lim sup ViPY n n
o
=lim p= ViPi n n
(3.10)
because for such a function, one immediately obtains that R = + CQso this function admits its analytic extension in the whole complex band: B = ((0 + iv) E C/O E 0, v E (-co, +m)}. Going back on the function f = C - ia from Theorem 1, where a=ICI(ncosP+bsinP)
(3.11)
if we note c+(t) the analytic extension of C(0) and u(t) the analytic extension of a(O), it is clear that k(t) - ia is the extension of f(0). Now, based on Theorems 1 and 2, we can infer that, if the functions c( ), a( ) and also i’( ) satisfy the relation (3.11, the minimal surface exists at least in the vicinity of the curve (C). The domain D, where the minimal surface is defined, must be taken as the maximal rectangular domain which is common, in the sense of Theorem 2, to all the functions cl( ), c2( ), c3( ), a,( ), u2( ), u3( ). It is easy to see that there is a large class of functions (for example, the elementary functions) which all satisfy the relation (3.1). However, it takes place: u=
c X(i’XC) jc xi;/
cosP+
CXi;
ICI~
Ic XEl
Based on the formulas (3.2a), representation of the surface:
sin P
(3.2b) and (24,
(3.12) one gives, at least in the domain
D, the next
316
V. Visarion, M. Baubec ! Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
n(O, v) = S,(O, v) + S,(O, v) S,(O, v) = c(0) - $ S,(O, lJ>=+z(@)
P(0)
(3.13) +$
-
P(0)
- ...
+$‘“(e)
(3.14)
- ***
(3.15)
4. Conditions for the successive derivatives In this section, we agree that whenever the indetermination 0’ appears, we take 1 as its value. Also, when ‘a’ and ‘b’ are complicated arguments, we use the expression p[a, b] instead of ub. Instead of eb, we use exp[b]. It is really hard to find c( ) and P( ) so that both c( ) and a( ) should satisfy in an obvious manner (3.1). So, in this section we shall give some properties of the functions satisfying (3.1) and we shall obtain a significant class of functions c( ) and P( ), for which c( ) and a( ) satisfy (3.1). Let us consider the crowd of all functions f : I ---*R, f~ C”(I), for which there are a, cq p 20, a, (Y,/3 < +w, (a,) C R’ so that If’“‘] G @“p[n,
na,]
Vn 3 1
also
If] s (Y
and
(4.1) lip sup p[n, a, - l]
We note this crowd with L( 0). One checks that a function belonging to L( 0) satisfies (3.1). We shall verify further on that sums, products and some compositions of elements from L(O) remain in L( 0). Also, we note with LO(O), the subset of functions belonging to L(O), for which u = 0. Moreover, if one can take a, = 0, V n > 0, we shall note f E Ll( 0). Obviously, L l(0) C LO(O) C L( 0). One sees that a function belonging to LO(O), can be extended in the complex band B, defined previously. Further on, we shall prove four lemmas. LEMMA
1. Let us take f, g E L(0) (LO(O) or Ll(0)).
Then f +g E L(0) (LO(O) or Ll(0)).
PROOF. fEL(O)*3%p,a*O,
@,)C~,
so that
If’“‘] < c+“p[n, IZ a,] gEL(I)*~q,
with
P,,baO,
lip sup p[n, a, - I] 6 a < +a, (b,,)CR,
so that Ig@‘)J
b,]
with
li~supp[n,b,--l]sb<+m
CT= max{q
P = max(& P,>
then
where
obvious
q}
c, = maxia,,
b,]
V. Visarion, M. Baubec I Comput. Methods Appl. Mech.
Engrg. 141 (1997) 311-333
317
lip sup p[n, c, - l] s max{lim sup p[n, a, - 11, lim sup p[n, b, - 11) 6max{a,b}<+m so f+gEL(U). Also, it is obvious that if a = b = 0 - f, g E LO(U)-it results f + g E LO(U), and f, g E Ll([l) involves f + g E Ll(0). LEMMA
2. Let us take f, g E L(U) (LO(U) or Ll(U)). Then f. g E L(U) (LO(O) or Ll(0)).
PROOF.
So
If ‘“‘Iss (@“pin,
n
a,P,q,P,,a,b~O
4
k’“‘ls QPdn, n &I
(4, @,I c R
lim sup p[n, a, - l] G a < +m
lip sup p[n, b, - l] s b < +m
We use the formula
Majorizing
successively, we can write
](fg)‘“‘] $$
p[n -
C:lf’“‘l k’“-“I=
k, (n - k)b,_,]
s aa1 i
4~,P1-”
kzO C,kc&[k
k
CEfikBT-kp[n,
k uk + (n - k)b,_,]
k=O
=
ffn,p’I
i k=O
CEykp[n,
k
uk
+
+
-
k)b,-,I
k=O
where d, = max{b,, b,, . . . , b,}
c, = max{u,, a,, . . . , a,} u, = max{c,, d,}
Y = PIP1
(we
can assume, without reducing the generalisation, that PI > 0). It remains to prove that lim, sup p[n, u, - 11< +m. We have lip sup p[n, u, - l] S max{lip sup p[n, c, - 11,li,m sup p[n, d, - l]}
We estimate lim sup p[n, c, - l] (and similarly lim sup p[n, d, - 11). Let e > 0 be fixed, then 3 n, EN* so that p[n,u,--I]
and
p[n,b,-l]Cb+E
Vn?=nn,
We have pin, c, - II= max{p[n, a0 - 11, pin, a, - 11, . . . , p[n, a, - 11) and for n 2 n8, it has place: p[n, c, - 11s max{p[n, a, - I], p[n, a, - 11, . . . , pin, u,~_~ - I], u + E}
(4.2)
318
V. Visarion, M. Baubec
One observes now that a,, a,, . . + 3ane_, < 1. Indeed,
I Comput. Methods Appl. Mech. Engrg.
for E > 0 hxed, let CY,be
therefore
n,
141 (1997) 311-333
being
fixed,
we
can
assume
that
and then we have
5zni2 VnE{l,2,. If (9 =saa,p” d acw,p jf'"'l ~aqpnp[n,na,]for IZ3nE
..,nE -1)
In other words, increasing cy, we can choose a, = a, = . . . = a, E_, = 0.5 < 1 Of course, this artifice makes (Y dependent on E, but fixing E, (Y becomes fixed too. We observe however, that if a < 1 (particularly a = 0), then the artifice can be applied without making (Ydependent on E. Indeed, let a < 1 be. Then, there is n, E N* so that a, < 1, V n 3 n,, otherwise it results immediately that lip sup p[n, a, - l] 3 1 ,
contradiction.
Therefore, IZ, replaces nE in the artifice already used. Going back on the formula (4.2). Due to a,-1~0, ViE{O,l,...,n,-1}, there is n,EN*, therefore p[n,c,-l]
so thatp[n,u,-l]
Vnan,,
Vn3max{n,,n,}
From the last relation,
it results in
li,m sup p[n, c, - l]
I(fg)'"'l~(II(&)cuI(&)(P + P,)"A4n4J where li,m sup
p[n,
u, -
l] < max{u, b} + E
therefore f *g E L(U). If a = b = 0, then cr, czr do not depend on E, we make c-+0, and one obtains f *g E LO(O). Similarly, if f,g E Ll(O), then f.g E Ll(0). LEMMA
3. Let US take g E Ll(U), f~
Jg’“‘l s c@”
0, J intervals, 9 2 g(0) so
~f(n)~erJ?~p[n,nu,], Vnal
lip sup p[n, a, - l]
L(J),
a,P,q,Pr,a~O
(a,)Crw
f 0 g E L(0).
PROOF.
One can prove without difficulties the next formula (through
mathematical
induction,
for
example) :
(f0 g)‘“’
c
ZkZl f(k)
6;l,,,Lkg(il).
* . g(Q
i,+...+ik=n
i
i, P 1
where n 2 1 and by,..+, E N. Hence, it results in
(4.3)
V. Visarion, M. Baubec
I Comput. Methods Appl. Mech. Engrg.
141 (1997) 311-333
319
(4.4)
where the coefficients a: =
c
;,+...,k=rl
are generated
b;,...,,,
k=l,n,
using the recurrent
af: = u;I; + ka,k_, ,
ns2,
formula k=l,n, uf: = 0, for k = 0 or k = n + 1.
beginning with ai = 1, and by convention From (4.4), we shall have
s q(@)n
c p[k k u,]“,”
k=l
where r = max{ 1, cr&} . Because f E L(Q), there is b > 0 (b 3 a) such that p[k,kuk]<(bk)k,
Vksl
We use now, the Stirling’s formula: k! = kkePka
exp[0,/(12k)]
0, E (0,l)
with
C R
One obtains p[k, k a,] s bkek &
therefore
k! exp[-@/(12k)]
,
p[k, k a,] s (be)kk! so, it takes place:
j(fCl g)‘“‘l s a,($)”
2
(be)kk! uf:
k=l
s q($r,)”
k$l k! af:
rl
=
Let us take b,k = k! ui. From the recurrent V n 3 2 and further on: 3, ef i:
b,k = i
k=l
n-1
- 1) c
b;_,
k=l
therefore
$, 6 (2n - l)s”_,
and it results:
s, C 3 5 7.. . (2n - 1) = ,J’:(,“,‘), One checks that li,m sup Es2 n and putting
e
formula for at, one obtains for bf: : bf: = k(bil:
+ b,k_,) = I?; (2k + l)b:_,
k(b;I:
k=l
s(2n
max{ 1, be}
(4.5) + bt_,),
320
V. Visarion, M. Baubec I Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
p[n,nu,]= 3, = i:
k=l
k!
u,k )
we have
li,m sup p[n, U, - l] Sa Together
with the relation (43,
LEMMA
4. Let us take g E LlQ),
If(n)l~(ylPI,Ig(n)l~apn, then f0
this proves that f0 f E Ll(J),
VnaO,
g E L(0).
0, J intervals, J 2 g(0) so
and
a,p,a,,/31~0
and
r=max{l,@r)
g E LO(O).
PROOF.
We write as in previous lemma:
with a; = 1
izf:= u,“:; + k&
,
ns2
We note
s,=iuf: k=l
To prove the lemma it is necessary and sufficient to verify that li,m sup K, n Without difficulties (through mathematical
induction) one proves that
k (4.6) u~=~k”-(k-11)!1!(k-1)“+“‘+(-1)k2!(k!2)!2”+(-1)k+1 l!(k?l)! ‘”
and it results ’ + (-l)“-k
(4.7)
(n ! k)!
Obviously, S, 6 e kz, 6
Let us take T,, zfkgI g Let us prove that limE_=O n n One uses the next Property (which will be proved after the demonstration ‘If R : N* ---, N* is given such that R(n) 6 n, V n E N*, then:’
of this lemma):
V. Visarion, M. Baubec I Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
R(n)
“pnql@o! = O this property, we define
Admitting
‘thesmaZlest’mEl,n
np”
-m I
-
z~--,VpEl,n P!
.
It is clear now that
(R(n))”
Tns n (R(n))!
and
E,,
R(n)
4@m
n
how %i”-1
and with the Property), it results in
(in accordance
the Property:
We shall demonstrate PROPERTY.
Let us take R : N*+N*,
R(n)
“p n-(/m! PROOF.
=
R(n)sn,
Vn EN*.
Then:
O
We assume that
R(n) % = rq@(qj! do not converge to 0, for n+ 03, therefore,
3 (nk) C N* so that
wnk) nkvm!Ap>o
Obviously,
R(n,)
u “k
S--=1
-
then
7
p 4ip
R(nk),
inf -
nk
‘k
and 3 (n;) C (nk) so that lim
k
R(4) 4
-=yYp>o
but then, for E > 0, little enough, fixed and for k 2 k,, we have R(4)
->s-&>O+R(n;)2(y-&)n; n;
3 [(y - &)n;] * ‘qm
where we noted [xl, the integer part of the real number X. It is known that m&+03, therefore
forcz>O,
3 nqw
321
322
V. Visarion, M. Baubec
I Comput. MethodF Appl. Mech. Engrg.
141 (1997) 311-333
so, we shall have u
“k
I
=.
but how (u,;) C (u,,), supposition was false.
we have simultaneous Q.E.D.
uni, k\
0 and u,,~ - k
p > 0, contradiction,
then our
Now, we can prove the next result: THEOREM 3. Let (C) be a curve given through the function c : O+ R3, which satisfies the conditions (a), (b), (4 and 1e t us take P : 0-t R, PE C”(0). We assume that c( ), P( ) E Ll(0). Then, there is a continuous minimal surface in isometric coordinates which satisfies the conditions (1) + (2), defined at least in the vicinity of the curve. PROOF.
We take 9 = [rI, r2] C R, rl > 0,
d,q:O~[W3,g,1:O~[W,hl,h2:cD~[W,
functions defined thus d=Cx(i;xc)
q=cxi; h,(x) =x-l’*
l=lc1*=(&c)
g = jc x i;y = (c xi, h*(x) =x+l’*,
c x i;)
VxEJ
In accordance with the condition (c), let us take rl > 0 so that rt ~g(@), rl c l(O), V 0 E 0. That is why, let J = [rl , r2] be the domain of definition for h, and h,, r2 chosen so that J > g( 0) and J > l( 0). One verifies immediately that on an interval J = [rl, r2] with rl > 0, any power function y = xr, r E [w, belongs to L(J), therefore the functions h,, h, belong to L(J). It is now clear that C( ) E Ll(0). We have
c X(i’XC)
a=
Icxi;(
cosP+
. .. (CIGsinP
= d(h, 0 g) cos P + q(h, 0 l)(h, 0 g) sin P We have d, q, g, 1 E Ll( 0) {being sums and products of functions belonging to L l( 0)}, h, 0 g, h, 0 1 E L(U) {applying Lemma 3}, cos P, sin P E LO(U) {in accordance with Lemma 4}, and finally Q E L(O), {obviously, in accordance with Lemmas 1 and 2). Also, C E Ll(0) C L(0). It results in C, a satisfying the relation (3.1), therefore, in accordance with Theorem 2, they can be analytically extended in a rectangular domain D so that 0C D c @. Let these be extensions k(t), respectively, u(t). It is clear and in accordance with that f(O) = c(O) - ia(O) can be analytically extended in D through i(t)-ia Theorem 1, the desired minimal surface exists. Having continuous functions defined in such a domain D, the surface is continuous. It is obvious in the next corollary: COROLLARY.
Zf the curve (C)
is given through c : Iw+ R3 which satisfies the condition (b), also P : R+R and q( 1, c2( ), c3( 1, P( ) are trigonometrical polynomials, then the continuous minimal surface satisfying the conditions (1) and (2) exists at least in the vicinity of the curve. Indeed, a trigonometrical polynomial satisfies (a) and because the function c is periodical, (a) and (b) involve (c). Secondly, one verifies easily that a trigonometrical polynomial belongs to L l( R).
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323
The four lemmas and Theorem 3 give us only few sufJicient conditions for the existence of the surface, which is defined in a rectangular domain ID,= ((0, V) E R2/0 E 0, v f (-c, G)}, V > 0. The surface is represented in b through the series (3.13)-(3.15). We note that for a peculiar given case, we can try decomposing the functions c, a into sums, products and compositions of elementary functions and using the four lemmas proved forward, to find a domain D of definition of the minimal surface. Since the four lemmas use inequalities which can be still refined, the number of decompositions must be as little as possible, such a decomposition not being unique. Anyway, we shall not obtain, except for the simplest cases, the maximal domain of the minimal surface. The results proved until now, allow us to construct and to represent on a computer the minimal surface, at least in the domain b found before, for each special case. One notices that the hypothesis of the periodicity of the functions c( ) and P( ) is not necessary in the demonstration. Now, we shall look to some strange situations which take place. If c( ) is not periodical, the curve (C) is open, but of course a minimal surface which satisfies our conditions can pass through this curve. For example, when c( ) is periodical (of period Tc > 0) and P( ) is also periodical (of period Tp = n . Tc, n E N, n 2 2), the generated minimal surface looks like a cabbage, with ‘n’ leaves all passing through the curve (C), but which are continuously generated from one another, without cuts or discontinuities. If the ratio Tp/ Tc is not a rational number, or P( ) is not periodical and it is possible to extend 0, 0 = (-cc, +a), then the number ‘of the leaves of cabbage’ will be infinite, but countable. In this last case (Tp/Tc not rational), if we take 0 = [a, b], as a bounded interval and Tc = b - a, the surface will have a cut (a discontinuity), which will start right from the point c(u).
5. Representations
through
Fourier series
We shall give an alternating manner to write the solution x =x(0, V) in the case when c( ) and P( ) are periodical. To simplify, we assume 0 = R, Tp = Tc = 27r, c and a satisfying for example the conditions from Theorem 3. We notice that even if 0 = R, the images of the curve and of the surface can be obtained using only an interval of length 27~. Then, c( ) and a( ) can be decomposed in Fourier series, uniform convergent, thus ~(0) = 2 + 2 (cf, cos(m@) + c”, sin(m@)) ??I%1
(5.1)
a(O) = 3 + C (uf, cos(m@) + u”, sin@&)) VZZl
(5.2)
Based on the identities: cos(0 + iv) = cos(O)ch(v) sin(O + iv) = sin(O)&(v)
- i sin(O)sh(v)
(5.3a)
+ i cos(O)sh(v)
(5.3b)
formally replacing in the series (5.1) and (5.2), 0 with t = 0 + iv, replacing (5.1) and (5.2) in (2.5) and after integrating (2.5), one gets x(0, V) = 9 + C (CL cos(m@) + c”, sin(m@))ch(mv) ma1 +y
+ C A (a: cos(mO) + a”, sin(m@))sh(mv)
(5.4)
maI
We shall prove that this trigonometrical n given by Theorem 2, thus
representation
is equivalent with (3.13)-(3.15)
in the domain
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S,(O, v) = $ + ms, (ck cos(m@) + cS, sin(m@))ch(mv)
(5.5)
S,(O, v) = y
(5.6)
+ mT, t
(uf, co+&)
+ a: sin(m@))sh(mv)
Let us prove (5.5). We know that ch(mv)=l+02+04+ob+... 2!
4!
6!
We define
(mlJ)*‘-*
cmj dzf(cL cos(m@) + c”, sin(m@)) (2i _ 2)r It results immediately
V3 i 2 1
in:
$ + mz, ,gi cmj = 2 + 2 (cf, cos(m@) f c”, sin(m@))ch(mv) / _ mrl
?+cIx jP1
cmj = $?- + x
ma1
( C (cf, cos(m@) + c”, sin(mO))m”-*)
jr1
ma1 2
=
c(@)
-
7
(V)“-*
(5 - 2)!
4 p(@)
+
5
c(4y@)
-
. . .
;I
As we know from the theory of double series, for any U,j E R, m, i E N, if we have (A)
“I”,mzi
Iu,~( < +-cc,
Vja
1
and (B)
c$<+w jsl
it results in: (‘)
zl
_
,z
c
‘mj
=
,z
zl
‘mj
<
+cD.
Therefore, to prove (5.5), it is enough to verify (A) and (B) for the components remember that for an arbitrary function F( ), which satisfies F : [O, 27r] + R, F E C”([O, 2n])
of cmj. Now, we
F(0) = F(2T)
and F(Q)=$+nC1
3
(ff, cos(m@) + f”, sin(m@))
then the following formulas take place (here, the periodicity
of the function F is essential!).
(5.7) Let us prove (A), for example, for ckj where cmj = (ckj, cij, cRj). Then, for any j 3 1, we get (applying (5.7) for cr):
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< +a
therefore (A) is verified. It is easy to see that for v E (-R, +R) (where R is common in the sense of Theorem components ci( ), a,( ). i = 1,2, 3): C
(Y,-’
Ic;2j)l
j" 1
2, to all the
(21
2)! <+w7
so (B) takes place too. Therefore, (C) results for cf. Thus, we proved (5.5) in its first component. Similarly, one proves entirely (5.5) and further on (5.6), for any 0 E R and Y E (-R, +R), that is any (0, V) E 6. If c is a trigonometrical polynomial, from the two representation for x =x(0, Y), we see that the trigonometrical representation (5.4) is more convenient, because (5.5) becomes a finite sum. As the Fourier coefficients decrease extremely fast, one gets an excellent convergence which allows with a little number of terms, a good representation of the surface, in the domain b of uniform convergence on compact subsets. Up until now, we have studied the existence and the manner of generation of the surface somehow in the vicinity of the curve. What is the maximal domain of definition of the minimal surface? What is the behavior of the surface at the distance of the curve, in the case it exists? For some cases, we shall give an answer in the next section.
6. The case of a curve given through trigonometrical polynomials We want to find the maximal domain of definition of the generated minimal surface and some of its properties to some peculiar cases. Thus, starting from (2.5), we shall find the maximal analytic extension off(O), then we shall calculate the primitive u(t) off(t) by means of a complex integral, and then we shall find x = x(0, v). We shall rely on the next result: THEOREM 4. Let us define f : A + in a domain A c @, x(0, v) = re f(t), g:A*-+C
g(0
+ iv) zfR(O,
@, f(0 + iv) = x + iy, x = x(0, v), y = ~(0, V) an
analytic function
~(0, v) = im f(t) and further on, we take v) + iQ(@, V)
R(O, v) = re g(t), Q(@, V) = im g(t) where
Then,
R(@,zJ)=~~
Q(@,u)=sign(y)iy
(6.1)
A* = A\{(@ + iv)) E A/y(O,
v) = 0, x(0, v) s O}
(6.2)
g is an analytic function in A* and g2 = f.
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PROOF. On the one hand, one verifies that A* is an open subset in C. On the other hand, it is easy to prove that the functions: R, : (w2+Iw
Q, : Lv+[w
defined thus: R,(u,u)=~~
Q,(.,v)=sign(u)iT
(6.3)
are continuous and admit continuous partial derivatives on R2\{(u, 0) E R2/u SO}. It is clear in these conditions that R = R,(x(O, v), ~(0, v)) and Q = Q,(x(O, v), ~(0, v)) are continuous and admit for R and continuous partial derivatives in A*. Also, one verifies in A* the relations Cauchy-Riemann Q.E.D. Q, whence results in g being analytic. One checks immediately that g2 =f. We shall conclude that for ZJ= 0, u < 0, Q, is really discontinuous, l$nQ,(u,u)=+v’Z,
lim Q,(u, u) = -vG
U/10 u-0
thus
for a<0
u-0
Also, the function g’ : A* + C, g’ = -g fulfills the conditions of the previous see further on that the necessary additional condition to extend another positive us to use only the function g. Let us assume that c( ) and P( ) are trigonometrical polynomials of period study the behavior of any function which will appear, due to its periodicity, it research to the interval [0,2rr]. Therefore, we have . .. a=Cx(ExC) f=C-ia P cosP+ICI -sin c xc lb xi;1 Ic xc(
theorem, but we shall known function allows 27~. We notice that to is enough to limit the
Obviously, C, i x E, c x (i’ XC), IC x i:12, I&l2 a d ml‘t analytic extensions in the whole complex plane because they are trigonometrical polynomials so their Fourier developments have a finite number of terms. Also, using (5.3), the functions cos P and sin P can be extended in the whole complex plane. The functions IC x Cl and (Cl have a more delicate situation. Thus Jc X?l=VjGq
Ic]=m
We shall use the theorem
that we have proved previously.
Let us define
Xf, x,, Yr, Y, : R2-+ R X,(0, V) = re{) i Xi;12(0 + iv)}
X,(0, V) = re{ ICI’(0 + iv)}
Y,(O, V) = im{ IC X t12(0 + iv)}
YJO, V) = im{ ICI’(0 + iv)}
MI, = ((0 + iv) E @/X&O, V) G 0, Y&O, V) = 0) Ml, = ((0 + iv) E C/X,(@, V) G 0, Y,(O, V) = 0} Based on Theorem /C x i;l : D,+C
4, one defines the following where
It\ : ID,+ @ where
D, = C\M,
D, = C\M,
\C x i;l(O+ iv) = R(O, v) + iQ(@, v)
R = R,@-JO, ~1, y,(@, ~1)
R, Q : ~,-+rW
Q = Q,(X,(@, v), Yj@, ~1)
(6.4)
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I Comput. Methods Appl. Mech. Engrg.
ICl(O + iv) = R(O, V) + i&O, r~) R = ~,(X,(@
VI, yp,
v>)
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R, Q : Cl,+ R
C?= Q,(X,(@,
6,={(O,v)E[W’/(0+i~)E[0)~}
327
(6.5)
II), YJ@, ~1)
rr0, = ((0, V) E R*/(O + iv) E KD,}
Thus, it is clear that JC x i;l(@ + iv)\,=, = I(c x i;)(O)1
and
Ic[(@ + iv)l,=, = Ic(@)[
the right terms of these two equalities being the real functions. To simplify, we use the same notation for the extended function and for the real function, only the arguments being different. Now, we have all the elements to define the maximal analytic extension of a(O) andf(O) and let us take t = 0 + iv. We get u(t)
=
ccx (i’ x c))(t) (cos P)(t)
+ ICI(t) iz z $7
Ic x i;l(t)
(sin P)(t) (6.6a,b)
f(t) = d(t) - ia where u(t) and further on f(t)
are defined in
A = {all the simple connected domains D’ c c/R where D* = D, II D,. We notice that in some peculiar could be only special cases, when than D, n D,. We took D the biggest simply possibility to calculate a primitive surface. The condition R C D is generated surface. Integrating (6.6b), one obtains
C LCD’ C ID*}
(6.7)
cases, for example when P( ) = 0 we must take ID* = Dr. Also, there due to some simplifications appeared in (6.6a), the crowd D* is larger connected domain included in D*, which contains R, to have the u(t) off(t) in [IDand to find after that, x = x(0, v), which gives us the obviously necessary to establish the condition that (C) lies on the
u
: Ill-,
C3
@+iv
u(t) = c(t) - i
where we integrate
I 0
a(t) dt
in a way included in D. Finally, we get x : 64
(6.8) R3
x(0, V) = re(u(t)) = re(c(t)) + im (i@+iV u(t) dt) (6.9)
b = ((0, V) E R*/(O + iv) E D}
We saw that from R2 as the maximum possible domain of definition of the surface, at least the following crowd is excluded (excepting some special cases, as P( ) = 0, when only the first crowd from the following reunion is excluded): M = ((0, V) E lR’lX,(O, V) S 0, y,(O, V) = O} u ((0, V) E R”lX,(O,
V) SO, YJO, V) = O}
Even if lti x ii(t) is at the numerator in the formula (6.6a), one sees that the values ‘0, Y’ for which X,(0, V) = Y,(O, V) = 0, do not belong to D 1, so that these values must not be especially excluded. One notices that the discontinuity of Q, (for u 6 0, u = 0) will be transmitted to the function x, in the points of Ml, thus the surface, after a quiet and continuous zone in the vicinity of the curve (C)-that is, at least the zone where the series (3.13)-(3.14) and (5.4) converge uniformly on compact subsets-will be merely cut, being as a lot of bands (or petals) starting from the continuous zone of the surface, evolving then independent from one another.
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The uniform convergence of the series (3.14)-(3.15) and (5.4) fails when the cuts appear. Although being continuous surfaces, the petals have in the same time a freakish behavior, distance between them increasing in accordance with exponential laws. The equations of the cuts are {X,(0, V) c 0, Y&O, V) = O} and
the
(6.10)
{X,(0, V) c 0, YJO, V) = 0}
These systems of equations can be solved only in numerical ways, excepting some simple cases. For example, for curves like c,(O) = r cos(0) being a trigonometrical Xf = 1 + i
c*(O) = r sin(@)
)
polynomial
((1 + ml)(G,,
and ~~(0)
of ‘N’th degree, Eqs. (6.10) are written:
cos(m - 1)O + ZMIsin@ - I)O)ch(m - 1)~
I=1 m=l +
(1 - ml)(H,,
Yf= i
cos(m + I)@ + .Z,! sin@ + I)O)ch(m + Z)V)6 0
((l+mf)(-G,,sin(m-I)O+Z,,cos(m-I)O)sh(m-I)v
I=1 ??I=1 +
(1 - m/)(--H,,
(6.11)
sin@ + 1)O + .ZM1 cos(m + Z)O)sh(m + I)v) = 0
and similar expressions for Xg and Yp. Here, G,,, H,,, Zmr, .Zmr,m, 15 1, are constant coefficients and r E R. The equations of cuts can be directly solved in some simple cases. For example: Cl(O) = r cos(0)
(a)
)
~~(0) = r sin(@) ,
c3(0)=0
with
r>O
In this case, the curve is a circle of radius ‘r’. Then, (C x Cl(t) = r, Iii(t) = r and one sees that the equations of cuts have not solutions. In other words, through a circle of arbitrary radius pass an infinite lot of minimal surfaces+ne for each function P( ) trigonometrical polynomial-which are not cut and their domain of definition being ll?. Also, the series (3.14), (3.15) and (5.4) re p resent the full surface because one verifies that a( ) belongs to LO(O) so the series converge uniformly in any compact subset lEC R2. At any rate, such a surface must be not bounded for V+ +w or V+ --co or both, due to Liouville’s theorem applied to the harmonic functions xi = ~~(0, v), j = 1,3, defined in the entire complex plane. For P( ) = 0, the surface is the plane of the circle, while for P( ) = ct. # 0, the minimal surface is a catenoid (the only minimal surface of rotation). Of course, there are other minimal surfaces that pass through the circle, surfaces which can be cut, corresponding to a function P( ) expressed through a polynomial, does not trigonometrical infinite series. It is obvious that P( ), as a trigonometrical introduce cuts. We also notice that if ci( ), c2( ) are arbitrary and c3( ) = 0, P( ) = 0, the minimal surface is not cut being the plane of the curve. c,(O) = r cos(0)
(b)
where aC[W, NEN, Id X
e]‘(t)
= E
N>l, + D
,
r>O.
c,(O)=rsin(@),
c3(0) = a cos(NO)
So, we have
cos(2NO)ch(2Nv)
If_?l’(t)= C - (C - 1) sin(2NO)ch(2Nv)
- iD sin(2NO)sh(2Nv) - i(C - 1) cos(2NO)sh(2Nv)
E, D, C are positive constants,
depending on r, N, (Y and F = E/D. The solutions of the first system of cuts for 0 E [0,2n], will be straight semi-lines, perpendicular the real axis, more precise: ‘2N’ pairs of semi-lines, given by ((0, V) E R2/0 = O,, tr E (-m, -v~)
U (vf, +m)}
where
z+> 0
on
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329
with kn Ok=%,
k-odd ,
OGkC4N,
The solutions of the second system of cuts will be similar, straight semi-lines, - l)]:
more exactly [here
L = C/(C
‘2N’ pairs of semi-lines: ((0, V) E [w*/o = &’ I, E (-00, v,) U (vg, +m)}
where
vg> 0
with km A==+&
k-even ,
OSk<4N,
vk=&ln(L+VZ7)
Therefore, a surface generated by P( ) # 0, after the zone of continuity given by Y E (-vmin, ZJ,~,) where v,.,,~” = min{ 3, v~}, divides itself into ‘4N’ petals, both v > 0 and Y < 0. If P( ) = 0, therefore the curve is an asymptotic line on the surface, then the cuts generated by the second system vanish and remain only ‘2N’ petals. We manage to construct and to see few surfaces, but the petals are so whimsical that only by rotating the surface one understands in some cases, the spatial shape and the evolution of these petals. One sees from (6.6a) that if P( ) = 1r/2, so the curve is a geodesic line on the surface, then the cuts generated by both systems (6.10) are present. Por more general cases, the systems (6.10) or (6.11) are more complicated, but, noticing that-when c( ) is a trigonometrical polynomial-for 0 = ct, X,, Yr, Xg and Yp have the behavior of some polynomials in the variable x = e’, their coefficients of maximal degree depending by the coefficients of maximal degree of the trigonometrical polynomial c( ), one proves the next property of regularity of cuts. THEOREM 5. In the plane of the coordinates -0, V-, the cuts converge asymptotically (for u-+ km) to some straight lines perpendicular on the real axis. These lines depend only on the coefficients of implicitly the number of cutsmaximal degree from c1 ( ), c2( ) and c3( ), their number--and depending on the degrees of the trigonometrical polynomials c1 ( ), c2( ), c3( ).
The demonstration, even laborious from a calculus point of view, does not raise difficulties of principle. In the plane of the coordinates, the cuts (excepting the simple cases presented before) are curved lines and they must be found in a numerical way. They make difficulties at the determination of the way in which we calculate the integral (6.8), when the curve (C) is complicated. The only easily generating zone is the continuous one, through the formula (6.9). Also, we still have two ways to represent at least a part of this zone (the uniform convergent on compact subsets one), through the series (3.13)-(3.15) and (5.4). We do not know if these two zones are identical. From the graphical study, they seem to be. Let E = F and G = 0 be the coefficients of the first fundamental form of a generated minimal surface. In a point of the curve, c(O), (belonging to the surface too), due to (2.6a), we have E = lx,o1* = ~~(0)~‘3 (Y2 > 0. E = E(O, V) is a continuous function in aa (the domain of definition of the surface). Then, each point c(O) has a neighborhood 0 c D, such that E = E(O, V) > 0, V (0, V) E 0, therefore the minimal surface has no singular points in 0.
7. Conclusions
(1) For a curve satisfying some conditions, there is a non-countable lot of minimal surfaces, all passing through the curve. Such a surface has a prescribed tangent plane along the generating curve and it admits a system of global and isometric coordinates.
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(2) Excepting few particular cases, the surfaces given through trigonometrical polynomials have the tendency to divide themselves in petals, after a continuous zone in the vicinity of the curve. This is a sign of the difficulty to find a solution of Plateau’s problem in the sense of least area. (3) If the membrane (the film of soap) lying on a rigid curve given through a trigonometrical polynomial, admits a system of coordinates as we considered, it has the boundary conditions, namely the function P( ), given through trigonometrical series, which cannot be reduced at a polynomial, because otherwise, it cuts itself. There are some notable exceptions, as any plane curve, where the function P( ) = 0 generates the plane of the respective curve as a surface of least area. (4) The aspect of some surfaces, as those which we mentioned before, could be studied, based on a package of programs which make numerical calculus, plotting of the cuts in the plane of the coordinates (0, v), construction, plotting and rotation of the generated minimal surfaces in R3. (5) All the constructed minimal surfaces are solutions of Plateau’s problem in the sense of [l], the required continuous zone, without singular points being at least the reunion of some neighborhoods, by one around every point of the generating curve. (6) One sees that for the construction of a minimal surface as before, it is necessary and sufficient to exist the field of tangent vectors at the curve, denoted c, and the regular enough field of vectors along the curve, denoted a, such that a(@)-~ti(@) and la(@)1 = IC(O)l, V 0 E 0. If c and (I belong to L(O), then the existence of the minimal surface is assured. Generally speaking, we can define the field a, without using the Frenet’s frame of the curve. Therefore, the method can be used on curves for which the oneness of the Frenet’s frame is not assured. Thus, if c( ) give the parametric coordinates of a straight line-for example, choosing c,(O) = ~~(0) = 0, ~~(0) = Oand taking u,(O) = cos(cyO), u,(O) = sin(&), u,(O) = 0, CY# 0, one generates a right helicoid, which its central axis is the generating straight line.
8. Some indications on the figures On all the figures, the generating curve is green. In all cases, we drew coordinate curves Y = ct. Also, we needed some time to complete few figures with curves 0 = ct. We represented the continuous zone of the minimal surface generated by the curve c,(O) = cos(O), c*(O) = sin(@), ~~(0) = 0.3 cos(20) and by P(0) = 0. The parameter v is varying between -0.48 and +0.48. One observes that exactly before the appearance of cuts, for IV] increasing, the coordinate curves I, = ct from the external zone, converge at a kind of spatial square, in the same time, the internal coordinates curves tending to a spatial cross with four arms. Generally speaking, if cg( ) is a trigonometrical polynomial of ‘N’th degree, the margin of the internal continuous zone is a cross with 2N arms and the margin of the external one is a polygon with ‘2N’ sides. Also, at least in the case of a curve on a cylinder and for a tixed function P, one can prove that if the generating curve ‘tends’ to be a circle, then the minimal surface ‘tends’ to be sagging, the continuous zone will be greater and the central cross tends to be more and more little and closed. At the limit, when the curve is a circle, the limits of the continuous zone tend to be infinite and the cuts vanish. Fig. l(a) presents this zone, seen from the axis 02, and Fig. l(b) offers a lateral view of the surface. For the previous curve and function P, one represents the starting portions of two adjoining external petals, in few successive positions. The parameter v is varying between -0.48 and -1.3. One sees how one petal ‘climbs’, in the same time, the other petal ‘descends’. The limits of the continuous zone are drawn with red. The other two undrawn petals are opposite and symmetrical given the drawn petal. The represented petals are fully generated for YE (-CQ, -0.48) and for 0 E (7r/4,31~/4), respectively (37r/4,51T/4). For the same curve and function P, we drew the generating curve v = 0 and the margins of all the four external petals (V < 0), through curves 0 = ct. We see from the axis 02 in Fig. 3(a) and from a ‘lateral point’ in Fig. 3(b). Minimal surface passing through the circle c,(O) = cos(O), ~~(0) = sin(O), c,(O) = 0, generated by P(0) = 0.45 cos(20). The parameter v is varying in the interval [0, +0.8]. As we know, such a surface
V. Visarion, M. Baubec
I Comput. Methods Appl. Mech. Engrg.
141 (1997) 311-333
(W
lb) F1g.1. Fig. 2.
(cl
331
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K Visarion, M. Baubec
I Comput. Methods Appl. Mech. Engrg.
(a)
141 (1997) 311-333
(a)
(b) Fig. 3. Fig. 4.
Fig. 5. Fig. 6.
V. Visarion, M. Baubec
I Comput. Methods Appl. Mech. Engrg.
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333
is not cut. The surface is seen from a ‘lateral point’ in Fig. 4(a) and from the plane of the circle in Fig. 4(b). Catenoid passing through the same circle, generated by the constant function P(0) = n/4. We plotted the cuts in the plane of coordinates (0, V) - 0 E [0,21r] and Y E [0,4]-, for the minimal surface generated by the curve c,(O) = cos(@) c?(O) = sin(@) c3(0) = 0.1 cos(0) + 0.1 sin(@) + 1.2 cos(20)
+ 0.5 sin(20)
+ + 0.02 cos(30)
+ 0.1 sin(30)
The cuts are white, the axis 00, OV and the lines 0 = 21r, v = 4 are red. One sees that due to the continuous zone, the cuts ‘start’ from a certain distance of the axis 00 and they tend to become parallel with the axis 0~. The points which cannot be linked with the real axis 00 through a continuous line which do not cross the cuts, must be excluded from the domain of definition of the surface, because the respective domain must contain the axis 00 and it must be simply connected. The cuts were found, solving in a numerical way Eqs. (6.10).
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll]
L. Bianchi, Lezioni Di Geometria Diferenziale (Enrico Spoelli, Pisa, 1902). K. Weierstrass, Mathematische Werke, Vols. l-7 (Edit. Mayer Berlin, 189441927). R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York-London, 1953). J. Nitsche, A new uniqueness theorem for minimal surfaces, Arch. Rat. Mech. Anal. 52(4) (1973). F. Tomi, On the local uniqueness of the problem of least area, Arch. Rat. Mech. Anal. 52(4) (1973). J. Nitscwlesungen iiber Minimalplachen (Springer, Berlin, 1975). V. Visarion and C. Stanescu, Calculul Starilor de Tensiune in Teoria Placilor Curbe (Le calcul des Ctats de tension dans la theorie des coques. Cylindres, surfaces d’aire minime) (Edit. Academic Roumaine, Bucharest, 1969). G.M. Fihtenholtz, Course of Intergral and Differential Calculus (Edit. Tehnica, Bucharest, 1969). R. Osserman, A Survey of Minimal Surfaces, 2nd edition (Dover Puublication, New York, 1986). C. Costa, Uniqueness of minimal surfaces embedded in R’ with total curvature 12rr, J. Diff. Geom. 30(3) (1989). D. Hoffman, Comment utiliser un ordinateur pour trouver de nouvelles surfaces minimales et des bulles de savon, University of Massachusetts, MA 01003.
ELSEVIER
Some strange properties of minimal surfaces in connection with Plateau’s problem Viorel Institute of Solid Mechanics,
Visarion*
, Margarit Baubec
Str. Constantin Mille, 1.5, 70701, Sect. 1, Boukarest,
Romania
Received 15 June 1995
Abstract In this paper we shall prove the existence of some minimal surfaces passing through enough smooth curves, surfaces which satisfy some given boundary conditions. We shall point out a strange property of ‘cutting’ of these surfaces, at least for large classes of curves and boundary conditions. The theoretical results will be completed, on the one hand, by a graphical study, to construct and to see the surfaces, and on the other hand, by a numerical study, to obtain the geometry of the cuts.
1. Introduction From geometrical point of view, a minimal surface is defined by the fact that in every point the mean curvature is null. In the sense of [l], Plateau’s problem consists in finding a portion of continuous minimal surface without interior singular points, terminated to a given and closed curve. The minimal surface 1ocuZZyminimizes (or maximizes) the area, among all the surfaces, passing through a local close contour lying on the minimal surface, which can be generated in a certain spatial vicinity of the minimal surface. Further on, we shall develop a method to generate continuous minimal surfaces passing through a given, fixed curve and we shall demonstrate accurately the existence of the solution, at least for large classes of curves and boundary conditions. In the case of a closed curve, if these minimal surfaces have not interior singular points, they are solutions of Plateau’s problem. For a class of curves still large enough, in which the parametric coordinates of the curve are given by trigonometrical polynomials, we shall bring into evidence an exotic property of ‘cutting’ of some surfaces, after a continuous zone lying in the vicinity of the curve. After this zone, the surface is ‘broken’ in a number of ‘petals’ evolving whimsically in the space. We completed this special study, in an accurately mathematical point of view, with the achievement of a package of programs, for graphics and numerical calculations, necessary on the one hand, to construct, to see and to rotate the studied surfaces and on the other hand, to solve on a numerical way the equations of ‘cuts’, which, as we shall see, cannot be solved directly except for the simplest cases. These programs were written in FORTRAN, calling however subroutines and functions of C language (especially for graphics).
* Corresponding
author.
0045-7825/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved PIZ SOO45-7825(96)01118-S
312
V. Visarion, M. Baubec
I Comput. Methods Appl. Mech. Engrg.
141 (1997) 311-333
2. Construction of minimal surfaces To simplify, we make the next convention. In few places in this paper, we say that a vectorial function satisfies a property (or a relation) which was defined for real functions. This will mean that the components of the given vectorial function satisfy the respective property. Let 0 c [w C Q= be an open interval and c : O+ R3, c(O) = (c,(O), c,(O), c3(0)), 0 E 0 a function which gives the parametric coordinates of a smooth enough curve (C) in R3, admitting in every point c(O), the existence and the oneness of F&et’s system of vectors. So naturally, the following conditions must be satisfied: (a) 4 )ECmQ) (c h as d erivatives of any order in 0). (b) The vectors {C(O), i;(O)} are independent linearly, for any 0 E 0, where we note C=a
dc
i;=- d2c de2
(c) 3 IY> 0 so that (YG IC(O)l and (Y< I(C x i)(O)1 V 0 E 0. This condition is not artificial, because one verifies immediately that (a) + (b) involve (c) in one of the following cases: - the function c( ) is periodical, with the period less than the length of the interval 0. - we restrict our theory at a compact interval J = [c, d] C 0. When (C) is a closed curve, these two cases take place in a natural manner. Let t, II, b : O+ R3, t = t(O), n = n(O), b = b(O) be Frknet’s system of vectors in a point c(O). It takes place: t=;
i x(i
XC)
n=IcIIcxtl
(i’xi’)
(2-l)
b= Ic xi;1
where lel = j/e: + e; + e: is the Euclidean norm of an arbitrary vector e. We try to find a minimal surface (S), defined in a domain D C_R2, conveniently chosen, given by x : 6+ lR3, x =x(0, v), in isometric coordinates, so that the following conditions should take place: (1) The curve iies on the surface, thus x(0,0) = c(O), V 0 E 0. (2) The tangent plane to the surface, in the points of the curve, must be established by the tangent t to the curve (or by C) and by a perpendicular vector to c, noted a, which must have the expression: a(@) = lti(@)l(n(@) cos P(O) + b(O) sin P(0))
V0 E 0
(2.2)
where the function P : O+ [w is at least continuous. It is obvious that a(@)lc(@) and la(@)1 = [C(O)/, V 0 E 0. Further on, we shall prove the following theorem: THEOREM
1. Let a, c, P, t, n, b be the real functions given before and f a complex function f : 0-+ C3:
f(0)~f~(@)-iu(O),
VOEO
where
i=fl
(2.3)
We assume that there is a simply connected domain ID and an analytic function, h : IID+-C3 so that UCD&_@ and hl,=f. Let DC[W2 be the domain: b={(0,v)EIW2/(0+iv)EDC@}. Then, there is an analytic function u : [ID+ C3, ri = h so that x = x(0, V) dzfre(u(8 + iv)), (0 + iv) E D, defines in isometric coordinates a minimal surface (S), furfilling (1) and (2) conditions. PROOF. Let us take u : ID-* C3 a primitive in D of h (established, x : !D+- lR3 x(0, v) = re(u(O + iv)).
excepting a complex constant)
and
V. Visarion, M. Baubec
We have, obviously, du -=p dt
I Comput. Methods Appl. Mech. Engrg.
using Cauchy-Riemann
relations:
d.x d.x d re(u) + i dim(u) ---jg--=~-i~=x,O d@
d re(u) + i dim(u) dt -= dt
313
141 (1997) 311-333
-ix,,
(2.4)
We know that for the surface (S), the vectors x,@ and x,, define the tangent plane at the surface in every regular point x(0, v). On the other hand, we have du = h(O) =f(O) dt ,,=,, OEO
= C(0) - iu(0)
(2.5)
From (2.4) and (2.5), it results in (2.6a,b) The relations (2.6a) and (2.6b) mean, in fact, that condition (2) is fulfilled. Also, integrating one obtains x(0,0)
(2.6a),
= c(0) + ct.
We remember now that ~(0 + iv) is established, excepting a complex vectorial constant, so x(0, V) is found up to a real vectorial constant. Conveniently fixing this constant, we can satisfy condition (1). ‘Now, let the analytic function g : III-C be defined thus: g(t) = h?(t) + h;(t) + h:(t)
(2.7)
We can write S(%EO =ff(@) =6:(o)
+fi(@)
+f:(@)
= (iI
- ia,(@
+ (&(O) - ia,(@
+ (C,(O) - ia,(0))2
+ c$(O) +6:(o)
- (a~(@) + a:(@) + a:(@)) - 2i(k,(@)u,(@)
+ k,(@)u,(@) + t,(O)u,(O))
= IC/*- la/* - 2i(d, u) = 0
(2.8)
where (C, u) is the scalar product of the vectors C, a. Due to the oneness of the analytic extension of a complex function in a non-empty &LEO = 0,
it results that
g(t)ltED
=
domain, from
0.
One obtains g(t)=(s)*+
(%)*+
(s)*=O
in I[D
(2.9)
From the last relation, in accordance with Weierstrass’s theorem of representation for analytic functions, it results that x(0, V) defines in b a minimal surface in isometric coordinates and finally, the theorem is completely proved.
3. The existence of minimal surfaces In this section we shall give sufficient conditions for the existence of analytic extension of a function, so that it should be possible to find some classes of functions c : I+- R3 and P :04 R for which the existence of minimal surfaces can be rigorously proved. We rely on the next result:
314
V. Visarion, M. Baubec
THEOREM
I Comput. Methods Appl. Mech. Engrg.
141 (1997) 311-333
2. Let the function p : O+ R, 0c R, p E C”(U) be such that
3a>O
so that
YiPi n
limsuo-
1
n
(3.1)
Ip’“‘(O)l =sup
QEO
Then, there are R >O and an analytic function (-R, R)}, so that gl, =p. Moreover, g is given by the next series: g(0
g : [E-C
where
[E= ((0
+ iv) E @/O E II, v E
+ iv) = g,(O, v) + ig,(O, v)
where for any (0 + iv) E IE:
g,(O, v) =p(O) g2(@,
v) = +
-;
p’2’(o)
p”‘(@) _
+$
p’4’(o)
- ***
$ p’3’(@) + $ p’y@)
(3.2a) - ...
(3.2b)
PROOF. We shall show that the last two series (formally for the time being) converge uniformly in any compact subset of a domain E conveniently chosen. Weierstrass’s criterion of uniform convergence shall be applied. One takes the following rectangular domain: vE(-R,R)}
E={(O+iv)E@/OEO,
(3.3)
where R=-
1
ez2.71..
a2e2
.
(3.4)
therefore, we take E = ((0, v) E R2/(0 + iv) E E}. Let us take KC E a compact subset, then there is V > 0 so that Kc ((0, v) E R2/0 E 0, VE[-ti,C]}
and
i
Because V < R, there is p > (Y so that 1 i=-----p’e’ From the relation
(3.1), it results that for E = (p - cr)/2 > 0 and for p = (I! + E, 3 n, E N* so that
Ip’“‘(O)l < pnnn, V 0 E 0, V 122 n,.
In these conditions, it is obvious that, except for a finite number of terms which have no importance, the series (3.2a) is absolutely minorate, for 0 E 0 and v E (-Z?, +R), by the series: fi+(2p)2$+(4p)4$+(6p)6$+
where Z? is the convergence R=
1 li,m supm
...
(3.5)
radius of the last series.
where
b, = (2np)‘2”’ -$q
so
1
R=-_>v p2e2
It results that (3.5) absolutely majorizes (3.2a) for 0 E 0 and v E [-6, C]. Now, all the more, as it is clear that the numerical series:
V. Visarion, M. Baubec I Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
315
-2
p+(*p)2g+(4p)4$+(6p)6g+*..
(34
is convergent, while the series (3.5) and also (3.2a) excepting a finite number of terms, are absolutely minorate by (3.6) in K. In accordance with Weierstrass’s criterion, (3.2a) converges uniformly in K and further on in any compact subset included in 5. Similarly, we have uniform convergence on compact subsets in E for (3.2b). Moreover, we can see that the series obtained through partial derivations however more, are of the same type with the initial series and they also converge uniformly on compact subsets in E. In accordance with a well-known theorem, it results that g, and g, are continuously differentiable any number of times in E and their derivatives are given by the partial derivatives of the respective order of the series (3.2a) and (3.2b) (which give us g, and gz). Further on, one sees that $
=
%I -=__-
p(e)
_
1”!
g
p’3’(@)
p(@)
+
+
g
$
p’(@)
-
#y@)
-
. . . =
. . . =
-
2
(3.7)
$
(3.8)
av
As we know, whether in a domain (E in our case) g, and g, have continuous partial derivatives of first order and they satisfy the relations (3.7) and (3.8) in E (named Cauchy-Riemann relations), then the function g : [E+ C, g(0 + iv) = g,(O, V) + ig,(O, V) is analytic in IE. Finally, one observes that and
g,(@, 0) = P(@)
g,(O,O) =0
VO E 0
(3.9)
so gl, =p. The domain IEand the function g being established, the demonstration is complete now. Essentially, we have proved the next result: ‘Any real function, defined on an interval of the real axis, having derivatives of any order, and that has successive derivatives that do not increase too fast, in the sense of relation (3.1), admits an analytic extension in the complex plane, at least in a symmetrical vicinity of the real interval.’ Also, we can prove this result, observing that such a function ‘p’ is analytic on 0 and using its Taylor series. A special interest presents a function p : O- R, for which (Y= 0, so lim sup ViPY n n
o
=lim p= ViPi n n
(3.10)
because for such a function, one immediately obtains that R = + CQso this function admits its analytic extension in the whole complex band: B = ((0 + iv) E C/O E 0, v E (-co, +m)}. Going back on the function f = C - ia from Theorem 1, where a=ICI(ncosP+bsinP)
(3.11)
if we note c+(t) the analytic extension of C(0) and u(t) the analytic extension of a(O), it is clear that k(t) - ia is the extension of f(0). Now, based on Theorems 1 and 2, we can infer that, if the functions c( ), a( ) and also i’( ) satisfy the relation (3.11, the minimal surface exists at least in the vicinity of the curve (C). The domain D, where the minimal surface is defined, must be taken as the maximal rectangular domain which is common, in the sense of Theorem 2, to all the functions cl( ), c2( ), c3( ), a,( ), u2( ), u3( ). It is easy to see that there is a large class of functions (for example, the elementary functions) which all satisfy the relation (3.1). However, it takes place: u=
c X(i’XC) jc xi;/
cosP+
CXi;
ICI~
Ic XEl
Based on the formulas (3.2a), representation of the surface:
sin P
(3.2b) and (24,
(3.12) one gives, at least in the domain
D, the next
316
V. Visarion, M. Baubec ! Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
n(O, v) = S,(O, v) + S,(O, v) S,(O, v) = c(0) - $ S,(O, lJ>=+z(@)
P(0)
(3.13) +$
-
P(0)
- ...
+$‘“(e)
(3.14)
- ***
(3.15)
4. Conditions for the successive derivatives In this section, we agree that whenever the indetermination 0’ appears, we take 1 as its value. Also, when ‘a’ and ‘b’ are complicated arguments, we use the expression p[a, b] instead of ub. Instead of eb, we use exp[b]. It is really hard to find c( ) and P( ) so that both c( ) and a( ) should satisfy in an obvious manner (3.1). So, in this section we shall give some properties of the functions satisfying (3.1) and we shall obtain a significant class of functions c( ) and P( ), for which c( ) and a( ) satisfy (3.1). Let us consider the crowd of all functions f : I ---*R, f~ C”(I), for which there are a, cq p 20, a, (Y,/3 < +w, (a,) C R’ so that If’“‘] G @“p[n,
na,]
Vn 3 1
also
If] s (Y
and
(4.1) lip sup p[n, a, - l]
We note this crowd with L( 0). One checks that a function belonging to L( 0) satisfies (3.1). We shall verify further on that sums, products and some compositions of elements from L(O) remain in L( 0). Also, we note with LO(O), the subset of functions belonging to L(O), for which u = 0. Moreover, if one can take a, = 0, V n > 0, we shall note f E Ll( 0). Obviously, L l(0) C LO(O) C L( 0). One sees that a function belonging to LO(O), can be extended in the complex band B, defined previously. Further on, we shall prove four lemmas. LEMMA
1. Let us take f, g E L(0) (LO(O) or Ll(0)).
Then f +g E L(0) (LO(O) or Ll(0)).
PROOF. fEL(O)*3%p,a*O,
@,)C~,
so that
If’“‘] < c+“p[n, IZ a,] gEL(I)*~q,
with
P,,baO,
lip sup p[n, a, - I] 6 a < +a, (b,,)CR,
so that Ig@‘)J
b,]
with
li~supp[n,b,--l]sb<+m
CT= max{q
P = max(& P,>
then
where
obvious
q}
c, = maxia,,
b,]
V. Visarion, M. Baubec I Comput. Methods Appl. Mech.
Engrg. 141 (1997) 311-333
317
lip sup p[n, c, - l] s max{lim sup p[n, a, - 11, lim sup p[n, b, - 11) 6max{a,b}<+m so f+gEL(U). Also, it is obvious that if a = b = 0 - f, g E LO(U)-it results f + g E LO(U), and f, g E Ll([l) involves f + g E Ll(0). LEMMA
2. Let us take f, g E L(U) (LO(U) or Ll(U)). Then f. g E L(U) (LO(O) or Ll(0)).
PROOF.
So
If ‘“‘Iss (@“pin,
n
a,P,q,P,,a,b~O
4
k’“‘ls QPdn, n &I
(4, @,I c R
lim sup p[n, a, - l] G a < +m
lip sup p[n, b, - l] s b < +m
We use the formula
Majorizing
successively, we can write
](fg)‘“‘] $$
p[n -
C:lf’“‘l k’“-“I=
k, (n - k)b,_,]
s aa1 i
4~,P1-”
kzO C,kc&[k
k
CEfikBT-kp[n,
k uk + (n - k)b,_,]
k=O
=
ffn,p’I
i k=O
CEykp[n,
k
uk
+
+
-
k)b,-,I
k=O
where d, = max{b,, b,, . . . , b,}
c, = max{u,, a,, . . . , a,} u, = max{c,, d,}
Y = PIP1
(we
can assume, without reducing the generalisation, that PI > 0). It remains to prove that lim, sup p[n, u, - 11< +m. We have lip sup p[n, u, - l] S max{lip sup p[n, c, - 11,li,m sup p[n, d, - l]}
We estimate lim sup p[n, c, - l] (and similarly lim sup p[n, d, - 11). Let e > 0 be fixed, then 3 n, EN* so that p[n,u,--I]
and
p[n,b,-l]Cb+E
Vn?=nn,
We have pin, c, - II= max{p[n, a0 - 11, pin, a, - 11, . . . , p[n, a, - 11) and for n 2 n8, it has place: p[n, c, - 11s max{p[n, a, - I], p[n, a, - 11, . . . , pin, u,~_~ - I], u + E}
(4.2)
318
V. Visarion, M. Baubec
One observes now that a,, a,, . . + 3ane_, < 1. Indeed,
I Comput. Methods Appl. Mech. Engrg.
for E > 0 hxed, let CY,be
therefore
n,
141 (1997) 311-333
being
fixed,
we
can
assume
that
and then we have
5zni2 VnE{l,2,. If (9 =saa,p” d acw,p jf'"'l ~aqpnp[n,na,]for IZ3nE
..,nE -1)
In other words, increasing cy, we can choose a, = a, = . . . = a, E_, = 0.5 < 1 Of course, this artifice makes (Y dependent on E, but fixing E, (Y becomes fixed too. We observe however, that if a < 1 (particularly a = 0), then the artifice can be applied without making (Ydependent on E. Indeed, let a < 1 be. Then, there is n, E N* so that a, < 1, V n 3 n,, otherwise it results immediately that lip sup p[n, a, - l] 3 1 ,
contradiction.
Therefore, IZ, replaces nE in the artifice already used. Going back on the formula (4.2). Due to a,-1~0, ViE{O,l,...,n,-1}, there is n,EN*, therefore p[n,c,-l]
so thatp[n,u,-l]
Vnan,,
Vn3max{n,,n,}
From the last relation,
it results in
li,m sup p[n, c, - l]
I(fg)'"'l~(II(&)cuI(&)(P + P,)"A4n4J where li,m sup
p[n,
u, -
l] < max{u, b} + E
therefore f *g E L(U). If a = b = 0, then cr, czr do not depend on E, we make c-+0, and one obtains f *g E LO(O). Similarly, if f,g E Ll(O), then f.g E Ll(0). LEMMA
3. Let US take g E Ll(U), f~
Jg’“‘l s c@”
0, J intervals, 9 2 g(0) so
~f(n)~erJ?~p[n,nu,], Vnal
lip sup p[n, a, - l]
L(J),
a,P,q,Pr,a~O
(a,)Crw
f 0 g E L(0).
PROOF.
One can prove without difficulties the next formula (through
mathematical
induction,
for
example) :
(f0 g)‘“’
c
ZkZl f(k)
6;l,,,Lkg(il).
* . g(Q
i,+...+ik=n
i
i, P 1
where n 2 1 and by,..+, E N. Hence, it results in
(4.3)
V. Visarion, M. Baubec
I Comput. Methods Appl. Mech. Engrg.
141 (1997) 311-333
319
(4.4)
where the coefficients a: =
c
;,+...,k=rl
are generated
b;,...,,,
k=l,n,
using the recurrent
af: = u;I; + ka,k_, ,
ns2,
formula k=l,n, uf: = 0, for k = 0 or k = n + 1.
beginning with ai = 1, and by convention From (4.4), we shall have
s q(@)n
c p[k k u,]“,”
k=l
where r = max{ 1, cr&} . Because f E L(Q), there is b > 0 (b 3 a) such that p[k,kuk]<(bk)k,
Vksl
We use now, the Stirling’s formula: k! = kkePka
exp[0,/(12k)]
0, E (0,l)
with
C R
One obtains p[k, k a,] s bkek &
therefore
k! exp[-@/(12k)]
,
p[k, k a,] s (be)kk! so, it takes place:
j(fCl g)‘“‘l s a,($)”
2
(be)kk! uf:
k=l
s q($r,)”
k$l k! af:
rl
=
Let us take b,k = k! ui. From the recurrent V n 3 2 and further on: 3, ef i:
b,k = i
k=l
n-1
- 1) c
b;_,
k=l
therefore
$, 6 (2n - l)s”_,
and it results:
s, C 3 5 7.. . (2n - 1) = ,J’:(,“,‘), One checks that li,m sup Es2 n and putting
e
formula for at, one obtains for bf: : bf: = k(bil:
+ b,k_,) = I?; (2k + l)b:_,
k(b;I:
k=l
s(2n
max{ 1, be}
(4.5) + bt_,),
320
V. Visarion, M. Baubec I Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
p[n,nu,]= 3, = i:
k=l
k!
u,k )
we have
li,m sup p[n, U, - l] Sa Together
with the relation (43,
LEMMA
4. Let us take g E LlQ),
If(n)l~(ylPI,Ig(n)l~apn, then f0
this proves that f0 f E Ll(J),
VnaO,
g E L(0).
0, J intervals, J 2 g(0) so
and
a,p,a,,/31~0
and
r=max{l,@r)
g E LO(O).
PROOF.
We write as in previous lemma:
with a; = 1
izf:= u,“:; + k&
,
ns2
We note
s,=iuf: k=l
To prove the lemma it is necessary and sufficient to verify that li,m sup K, n Without difficulties (through mathematical
induction) one proves that
k (4.6) u~=~k”-(k-11)!1!(k-1)“+“‘+(-1)k2!(k!2)!2”+(-1)k+1 l!(k?l)! ‘”
and it results ’ + (-l)“-k
(4.7)
(n ! k)!
Obviously, S, 6 e kz, 6
Let us take T,, zfkgI g Let us prove that limE_=O n n One uses the next Property (which will be proved after the demonstration ‘If R : N* ---, N* is given such that R(n) 6 n, V n E N*, then:’
of this lemma):
V. Visarion, M. Baubec I Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
R(n)
“pnql@o! = O this property, we define
Admitting
‘thesmaZlest’mEl,n
np”
-m I
-
z~--,VpEl,n P!
.
It is clear now that
(R(n))”
Tns n (R(n))!
and
E,,
R(n)
4@m
n
how %i”-1
and with the Property), it results in
(in accordance
the Property:
We shall demonstrate PROPERTY.
Let us take R : N*+N*,
R(n)
“p n-(/m! PROOF.
=
R(n)sn,
Vn EN*.
Then:
O
We assume that
R(n) % = rq@(qj! do not converge to 0, for n+ 03, therefore,
3 (nk) C N* so that
wnk) nkvm!Ap>o
Obviously,
R(n,)
u “k
S--=1
-
then
7
p 4ip
R(nk),
inf -
nk
‘k
and 3 (n;) C (nk) so that lim
k
R(4) 4
-=yYp>o
but then, for E > 0, little enough, fixed and for k 2 k,, we have R(4)
->s-&>O+R(n;)2(y-&)n; n;
3 [(y - &)n;] * ‘qm
where we noted [xl, the integer part of the real number X. It is known that m&+03, therefore
forcz>O,
3 nqw
321
322
V. Visarion, M. Baubec
I Comput. MethodF Appl. Mech. Engrg.
141 (1997) 311-333
so, we shall have u
“k
I
=.
but how (u,;) C (u,,), supposition was false.
we have simultaneous Q.E.D.
uni, k\
0 and u,,~ - k
p > 0, contradiction,
then our
Now, we can prove the next result: THEOREM 3. Let (C) be a curve given through the function c : O+ R3, which satisfies the conditions (a), (b), (4 and 1e t us take P : 0-t R, PE C”(0). We assume that c( ), P( ) E Ll(0). Then, there is a continuous minimal surface in isometric coordinates which satisfies the conditions (1) + (2), defined at least in the vicinity of the curve. PROOF.
We take 9 = [rI, r2] C R, rl > 0,
d,q:O~[W3,g,1:O~[W,hl,h2:cD~[W,
functions defined thus d=Cx(i;xc)
q=cxi; h,(x) =x-l’*
l=lc1*=(&c)
g = jc x i;y = (c xi, h*(x) =x+l’*,
c x i;)
VxEJ
In accordance with the condition (c), let us take rl > 0 so that rt ~g(@), rl c l(O), V 0 E 0. That is why, let J = [rl , r2] be the domain of definition for h, and h,, r2 chosen so that J > g( 0) and J > l( 0). One verifies immediately that on an interval J = [rl, r2] with rl > 0, any power function y = xr, r E [w, belongs to L(J), therefore the functions h,, h, belong to L(J). It is now clear that C( ) E Ll(0). We have
c X(i’XC)
a=
Icxi;(
cosP+
. .. (CIGsinP
= d(h, 0 g) cos P + q(h, 0 l)(h, 0 g) sin P We have d, q, g, 1 E Ll( 0) {being sums and products of functions belonging to L l( 0)}, h, 0 g, h, 0 1 E L(U) {applying Lemma 3}, cos P, sin P E LO(U) {in accordance with Lemma 4}, and finally Q E L(O), {obviously, in accordance with Lemmas 1 and 2). Also, C E Ll(0) C L(0). It results in C, a satisfying the relation (3.1), therefore, in accordance with Theorem 2, they can be analytically extended in a rectangular domain D so that 0C D c @. Let these be extensions k(t), respectively, u(t). It is clear and in accordance with that f(O) = c(O) - ia(O) can be analytically extended in D through i(t)-ia Theorem 1, the desired minimal surface exists. Having continuous functions defined in such a domain D, the surface is continuous. It is obvious in the next corollary: COROLLARY.
Zf the curve (C)
is given through c : Iw+ R3 which satisfies the condition (b), also P : R+R and q( 1, c2( ), c3( 1, P( ) are trigonometrical polynomials, then the continuous minimal surface satisfying the conditions (1) and (2) exists at least in the vicinity of the curve. Indeed, a trigonometrical polynomial satisfies (a) and because the function c is periodical, (a) and (b) involve (c). Secondly, one verifies easily that a trigonometrical polynomial belongs to L l( R).
V. Visarion, M. Baubec
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141 (1997) 311-333
323
The four lemmas and Theorem 3 give us only few sufJicient conditions for the existence of the surface, which is defined in a rectangular domain ID,= ((0, V) E R2/0 E 0, v f (-c, G)}, V > 0. The surface is represented in b through the series (3.13)-(3.15). We note that for a peculiar given case, we can try decomposing the functions c, a into sums, products and compositions of elementary functions and using the four lemmas proved forward, to find a domain D of definition of the minimal surface. Since the four lemmas use inequalities which can be still refined, the number of decompositions must be as little as possible, such a decomposition not being unique. Anyway, we shall not obtain, except for the simplest cases, the maximal domain of the minimal surface. The results proved until now, allow us to construct and to represent on a computer the minimal surface, at least in the domain b found before, for each special case. One notices that the hypothesis of the periodicity of the functions c( ) and P( ) is not necessary in the demonstration. Now, we shall look to some strange situations which take place. If c( ) is not periodical, the curve (C) is open, but of course a minimal surface which satisfies our conditions can pass through this curve. For example, when c( ) is periodical (of period Tc > 0) and P( ) is also periodical (of period Tp = n . Tc, n E N, n 2 2), the generated minimal surface looks like a cabbage, with ‘n’ leaves all passing through the curve (C), but which are continuously generated from one another, without cuts or discontinuities. If the ratio Tp/ Tc is not a rational number, or P( ) is not periodical and it is possible to extend 0, 0 = (-cc, +a), then the number ‘of the leaves of cabbage’ will be infinite, but countable. In this last case (Tp/Tc not rational), if we take 0 = [a, b], as a bounded interval and Tc = b - a, the surface will have a cut (a discontinuity), which will start right from the point c(u).
5. Representations
through
Fourier series
We shall give an alternating manner to write the solution x =x(0, V) in the case when c( ) and P( ) are periodical. To simplify, we assume 0 = R, Tp = Tc = 27r, c and a satisfying for example the conditions from Theorem 3. We notice that even if 0 = R, the images of the curve and of the surface can be obtained using only an interval of length 27~. Then, c( ) and a( ) can be decomposed in Fourier series, uniform convergent, thus ~(0) = 2 + 2 (cf, cos(m@) + c”, sin(m@)) ??I%1
(5.1)
a(O) = 3 + C (uf, cos(m@) + u”, sin@&)) VZZl
(5.2)
Based on the identities: cos(0 + iv) = cos(O)ch(v) sin(O + iv) = sin(O)&(v)
- i sin(O)sh(v)
(5.3a)
+ i cos(O)sh(v)
(5.3b)
formally replacing in the series (5.1) and (5.2), 0 with t = 0 + iv, replacing (5.1) and (5.2) in (2.5) and after integrating (2.5), one gets x(0, V) = 9 + C (CL cos(m@) + c”, sin(m@))ch(mv) ma1 +y
+ C A (a: cos(mO) + a”, sin(m@))sh(mv)
(5.4)
maI
We shall prove that this trigonometrical n given by Theorem 2, thus
representation
is equivalent with (3.13)-(3.15)
in the domain
324
V. Visarion, M. Baubec ! Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
S,(O, v) = $ + ms, (ck cos(m@) + cS, sin(m@))ch(mv)
(5.5)
S,(O, v) = y
(5.6)
+ mT, t
(uf, co+&)
+ a: sin(m@))sh(mv)
Let us prove (5.5). We know that ch(mv)=l+02+04+ob+... 2!
4!
6!
We define
(mlJ)*‘-*
cmj dzf(cL cos(m@) + c”, sin(m@)) (2i _ 2)r It results immediately
V3 i 2 1
in:
$ + mz, ,gi cmj = 2 + 2 (cf, cos(m@) f c”, sin(m@))ch(mv) / _ mrl
?+cIx jP1
cmj = $?- + x
ma1
( C (cf, cos(m@) + c”, sin(mO))m”-*)
jr1
ma1 2
=
c(@)
-
7
(V)“-*
(5 - 2)!
4 p(@)
+
5
c(4y@)
-
. . .
;I
As we know from the theory of double series, for any U,j E R, m, i E N, if we have (A)
“I”,mzi
Iu,~( < +-cc,
Vja
1
and (B)
c$<+w jsl
it results in: (‘)
zl
_
,z
c
‘mj
=
,z
zl
‘mj
<
+cD.
Therefore, to prove (5.5), it is enough to verify (A) and (B) for the components remember that for an arbitrary function F( ), which satisfies F : [O, 27r] + R, F E C”([O, 2n])
of cmj. Now, we
F(0) = F(2T)
and F(Q)=$+nC1
3
(ff, cos(m@) + f”, sin(m@))
then the following formulas take place (here, the periodicity
of the function F is essential!).
(5.7) Let us prove (A), for example, for ckj where cmj = (ckj, cij, cRj). Then, for any j 3 1, we get (applying (5.7) for cr):
V. Visarion, M. Baubec
=
47rIc;2’)l (2i _ 2)! 6
I Comput. Methods Appl. Mech. Engrg.
325
141 (1997) 311-333
< +a
therefore (A) is verified. It is easy to see that for v E (-R, +R) (where R is common in the sense of Theorem components ci( ), a,( ). i = 1,2, 3): C
(Y,-’
Ic;2j)l
j" 1
2, to all the
(21
2)! <+w7
so (B) takes place too. Therefore, (C) results for cf. Thus, we proved (5.5) in its first component. Similarly, one proves entirely (5.5) and further on (5.6), for any 0 E R and Y E (-R, +R), that is any (0, V) E 6. If c is a trigonometrical polynomial, from the two representation for x =x(0, Y), we see that the trigonometrical representation (5.4) is more convenient, because (5.5) becomes a finite sum. As the Fourier coefficients decrease extremely fast, one gets an excellent convergence which allows with a little number of terms, a good representation of the surface, in the domain b of uniform convergence on compact subsets. Up until now, we have studied the existence and the manner of generation of the surface somehow in the vicinity of the curve. What is the maximal domain of definition of the minimal surface? What is the behavior of the surface at the distance of the curve, in the case it exists? For some cases, we shall give an answer in the next section.
6. The case of a curve given through trigonometrical polynomials We want to find the maximal domain of definition of the generated minimal surface and some of its properties to some peculiar cases. Thus, starting from (2.5), we shall find the maximal analytic extension off(O), then we shall calculate the primitive u(t) off(t) by means of a complex integral, and then we shall find x = x(0, v). We shall rely on the next result: THEOREM 4. Let us define f : A + in a domain A c @, x(0, v) = re f(t), g:A*-+C
g(0
+ iv) zfR(O,
@, f(0 + iv) = x + iy, x = x(0, v), y = ~(0, V) an
analytic function
~(0, v) = im f(t) and further on, we take v) + iQ(@, V)
R(O, v) = re g(t), Q(@, V) = im g(t) where
Then,
R(@,zJ)=~~
Q(@,u)=sign(y)iy
(6.1)
A* = A\{(@ + iv)) E A/y(O,
v) = 0, x(0, v) s O}
(6.2)
g is an analytic function in A* and g2 = f.
V. Visarion, M. Baubec
326
I Comput. Methods Appl. Mech. Engrg. 141 (1997) 311-333
PROOF. On the one hand, one verifies that A* is an open subset in C. On the other hand, it is easy to prove that the functions: R, : (w2+Iw
Q, : Lv+[w
defined thus: R,(u,u)=~~
Q,(.,v)=sign(u)iT
(6.3)
are continuous and admit continuous partial derivatives on R2\{(u, 0) E R2/u SO}. It is clear in these conditions that R = R,(x(O, v), ~(0, v)) and Q = Q,(x(O, v), ~(0, v)) are continuous and admit for R and continuous partial derivatives in A*. Also, one verifies in A* the relations Cauchy-Riemann Q.E.D. Q, whence results in g being analytic. One checks immediately that g2 =f. We shall conclude that for ZJ= 0, u < 0, Q, is really discontinuous, l$nQ,(u,u)=+v’Z,
lim Q,(u, u) = -vG
U/10 u-0
thus
for a<0
u-0
Also, the function g’ : A* + C, g’ = -g fulfills the conditions of the previous see further on that the necessary additional condition to extend another positive us to use only the function g. Let us assume that c( ) and P( ) are trigonometrical polynomials of period study the behavior of any function which will appear, due to its periodicity, it research to the interval [0,2rr]. Therefore, we have . .. a=Cx(ExC) f=C-ia P cosP+ICI -sin c xc lb xi;1 Ic xc(
theorem, but we shall known function allows 27~. We notice that to is enough to limit the
Obviously, C, i x E, c x (i’ XC), IC x i:12, I&l2 a d ml‘t analytic extensions in the whole complex plane because they are trigonometrical polynomials so their Fourier developments have a finite number of terms. Also, using (5.3), the functions cos P and sin P can be extended in the whole complex plane. The functions IC x Cl and (Cl have a more delicate situation. Thus Jc X?l=VjGq
Ic]=m
We shall use the theorem
that we have proved previously.
Let us define
Xf, x,, Yr, Y, : R2-+ R X,(0, V) = re{) i Xi;12(0 + iv)}
X,(0, V) = re{ ICI’(0 + iv)}
Y,(O, V) = im{ IC X t12(0 + iv)}
YJO, V) = im{ ICI’(0 + iv)}
MI, = ((0 + iv) E @/X&O, V) G 0, Y&O, V) = 0) Ml, = ((0 + iv) E C/X,(@, V) G 0, Y,(O, V) = 0} Based on Theorem /C x i;l : D,+C
4, one defines the following where
It\ : ID,+ @ where
D, = C\M,
D, = C\M,
\C x i;l(O+ iv) = R(O, v) + iQ(@, v)
R = R,@-JO, ~1, y,(@, ~1)
R, Q : ~,-+rW
Q = Q,(X,(@, v), Yj@, ~1)
(6.4)
V. Visarion. M. Baubec
I Comput. Methods Appl. Mech. Engrg.
ICl(O + iv) = R(O, V) + i&O, r~) R = ~,(X,(@
VI, yp,
v>)
141 (1997) 311-333
R, Q : Cl,+ R
C?= Q,(X,(@,
6,={(O,v)E[W’/(0+i~)E[0)~}
327
(6.5)
II), YJ@, ~1)
rr0, = ((0, V) E R*/(O + iv) E KD,}
Thus, it is clear that JC x i;l(@ + iv)\,=, = I(c x i;)(O)1
and
Ic[(@ + iv)l,=, = Ic(@)[
the right terms of these two equalities being the real functions. To simplify, we use the same notation for the extended function and for the real function, only the arguments being different. Now, we have all the elements to define the maximal analytic extension of a(O) andf(O) and let us take t = 0 + iv. We get u(t)
=
ccx (i’ x c))(t) (cos P)(t)
+ ICI(t) iz z $7
Ic x i;l(t)
(sin P)(t) (6.6a,b)
f(t) = d(t) - ia where u(t) and further on f(t)
are defined in
A = {all the simple connected domains D’ c c/R where D* = D, II D,. We notice that in some peculiar could be only special cases, when than D, n D,. We took D the biggest simply possibility to calculate a primitive surface. The condition R C D is generated surface. Integrating (6.6b), one obtains
C LCD’ C ID*}
(6.7)
cases, for example when P( ) = 0 we must take ID* = Dr. Also, there due to some simplifications appeared in (6.6a), the crowd D* is larger connected domain included in D*, which contains R, to have the u(t) off(t) in [IDand to find after that, x = x(0, v), which gives us the obviously necessary to establish the condition that (C) lies on the
u
: Ill-,
C3
@+iv
u(t) = c(t) - i
where we integrate
I 0
a(t) dt
in a way included in D. Finally, we get x : 64
(6.8) R3
x(0, V) = re(u(t)) = re(c(t)) + im (i@+iV u(t) dt) (6.9)
b = ((0, V) E R*/(O + iv) E D}
We saw that from R2 as the maximum possible domain of definition of the surface, at least the following crowd is excluded (excepting some special cases, as P( ) = 0, when only the first crowd from the following reunion is excluded): M = ((0, V) E lR’lX,(O, V) S 0, y,(O, V) = O} u ((0, V) E R”lX,(O,
V) SO, YJO, V) = O}
Even if lti x ii(t) is at the numerator in the formula (6.6a), one sees that the values ‘0, Y’ for which X,(0, V) = Y,(O, V) = 0, do not belong to D 1, so that these values must not be especially excluded. One notices that the discontinuity of Q, (for u 6 0, u = 0) will be transmitted to the function x, in the points of Ml, thus the surface, after a quiet and continuous zone in the vicinity of the curve (C)-that is, at least the zone where the series (3.13)-(3.14) and (5.4) converge uniformly on compact subsets-will be merely cut, being as a lot of bands (or petals) starting from the continuous zone of the surface, evolving then independent from one another.
328
V. Visarion, M. Baubec
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The uniform convergence of the series (3.14)-(3.15) and (5.4) fails when the cuts appear. Although being continuous surfaces, the petals have in the same time a freakish behavior, distance between them increasing in accordance with exponential laws. The equations of the cuts are {X,(0, V) c 0, Y&O, V) = O} and
the
(6.10)
{X,(0, V) c 0, YJO, V) = 0}
These systems of equations can be solved only in numerical ways, excepting some simple cases. For example, for curves like c,(O) = r cos(0) being a trigonometrical Xf = 1 + i
c*(O) = r sin(@)
)
polynomial
((1 + ml)(G,,
and ~~(0)
of ‘N’th degree, Eqs. (6.10) are written:
cos(m - 1)O + ZMIsin@ - I)O)ch(m - 1)~
I=1 m=l +
(1 - ml)(H,,
Yf= i
cos(m + I)@ + .Z,! sin@ + I)O)ch(m + Z)V)6 0
((l+mf)(-G,,sin(m-I)O+Z,,cos(m-I)O)sh(m-I)v
I=1 ??I=1 +
(1 - m/)(--H,,
(6.11)
sin@ + 1)O + .ZM1 cos(m + Z)O)sh(m + I)v) = 0
and similar expressions for Xg and Yp. Here, G,,, H,,, Zmr, .Zmr,m, 15 1, are constant coefficients and r E R. The equations of cuts can be directly solved in some simple cases. For example: Cl(O) = r cos(0)
(a)
)
~~(0) = r sin(@) ,
c3(0)=0
with
r>O
In this case, the curve is a circle of radius ‘r’. Then, (C x Cl(t) = r, Iii(t) = r and one sees that the equations of cuts have not solutions. In other words, through a circle of arbitrary radius pass an infinite lot of minimal surfaces+ne for each function P( ) trigonometrical polynomial-which are not cut and their domain of definition being ll?. Also, the series (3.14), (3.15) and (5.4) re p resent the full surface because one verifies that a( ) belongs to LO(O) so the series converge uniformly in any compact subset lEC R2. At any rate, such a surface must be not bounded for V+ +w or V+ --co or both, due to Liouville’s theorem applied to the harmonic functions xi = ~~(0, v), j = 1,3, defined in the entire complex plane. For P( ) = 0, the surface is the plane of the circle, while for P( ) = ct. # 0, the minimal surface is a catenoid (the only minimal surface of rotation). Of course, there are other minimal surfaces that pass through the circle, surfaces which can be cut, corresponding to a function P( ) expressed through a polynomial, does not trigonometrical infinite series. It is obvious that P( ), as a trigonometrical introduce cuts. We also notice that if ci( ), c2( ) are arbitrary and c3( ) = 0, P( ) = 0, the minimal surface is not cut being the plane of the curve. c,(O) = r cos(0)
(b)
where aC[W, NEN, Id X
e]‘(t)
= E
N>l, + D
,
r>O.
c,(O)=rsin(@),
c3(0) = a cos(NO)
So, we have
cos(2NO)ch(2Nv)
If_?l’(t)= C - (C - 1) sin(2NO)ch(2Nv)
- iD sin(2NO)sh(2Nv) - i(C - 1) cos(2NO)sh(2Nv)
E, D, C are positive constants,
depending on r, N, (Y and F = E/D. The solutions of the first system of cuts for 0 E [0,2n], will be straight semi-lines, perpendicular the real axis, more precise: ‘2N’ pairs of semi-lines, given by ((0, V) E R2/0 = O,, tr E (-m, -v~)
U (vf, +m)}
where
z+> 0
on
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with kn Ok=%,
k-odd ,
OGkC4N,
The solutions of the second system of cuts will be similar, straight semi-lines, - l)]:
more exactly [here
L = C/(C
‘2N’ pairs of semi-lines: ((0, V) E [w*/o = &’ I, E (-00, v,) U (vg, +m)}
where
vg> 0
with km A==+&
k-even ,
OSk<4N,
vk=&ln(L+VZ7)
Therefore, a surface generated by P( ) # 0, after the zone of continuity given by Y E (-vmin, ZJ,~,) where v,.,,~” = min{ 3, v~}, divides itself into ‘4N’ petals, both v > 0 and Y < 0. If P( ) = 0, therefore the curve is an asymptotic line on the surface, then the cuts generated by the second system vanish and remain only ‘2N’ petals. We manage to construct and to see few surfaces, but the petals are so whimsical that only by rotating the surface one understands in some cases, the spatial shape and the evolution of these petals. One sees from (6.6a) that if P( ) = 1r/2, so the curve is a geodesic line on the surface, then the cuts generated by both systems (6.10) are present. Por more general cases, the systems (6.10) or (6.11) are more complicated, but, noticing that-when c( ) is a trigonometrical polynomial-for 0 = ct, X,, Yr, Xg and Yp have the behavior of some polynomials in the variable x = e’, their coefficients of maximal degree depending by the coefficients of maximal degree of the trigonometrical polynomial c( ), one proves the next property of regularity of cuts. THEOREM 5. In the plane of the coordinates -0, V-, the cuts converge asymptotically (for u-+ km) to some straight lines perpendicular on the real axis. These lines depend only on the coefficients of implicitly the number of cutsmaximal degree from c1 ( ), c2( ) and c3( ), their number--and depending on the degrees of the trigonometrical polynomials c1 ( ), c2( ), c3( ).
The demonstration, even laborious from a calculus point of view, does not raise difficulties of principle. In the plane of the coordinates, the cuts (excepting the simple cases presented before) are curved lines and they must be found in a numerical way. They make difficulties at the determination of the way in which we calculate the integral (6.8), when the curve (C) is complicated. The only easily generating zone is the continuous one, through the formula (6.9). Also, we still have two ways to represent at least a part of this zone (the uniform convergent on compact subsets one), through the series (3.13)-(3.15) and (5.4). We do not know if these two zones are identical. From the graphical study, they seem to be. Let E = F and G = 0 be the coefficients of the first fundamental form of a generated minimal surface. In a point of the curve, c(O), (belonging to the surface too), due to (2.6a), we have E = lx,o1* = ~~(0)~‘3 (Y2 > 0. E = E(O, V) is a continuous function in aa (the domain of definition of the surface). Then, each point c(O) has a neighborhood 0 c D, such that E = E(O, V) > 0, V (0, V) E 0, therefore the minimal surface has no singular points in 0.
7. Conclusions
(1) For a curve satisfying some conditions, there is a non-countable lot of minimal surfaces, all passing through the curve. Such a surface has a prescribed tangent plane along the generating curve and it admits a system of global and isometric coordinates.
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(2) Excepting few particular cases, the surfaces given through trigonometrical polynomials have the tendency to divide themselves in petals, after a continuous zone in the vicinity of the curve. This is a sign of the difficulty to find a solution of Plateau’s problem in the sense of least area. (3) If the membrane (the film of soap) lying on a rigid curve given through a trigonometrical polynomial, admits a system of coordinates as we considered, it has the boundary conditions, namely the function P( ), given through trigonometrical series, which cannot be reduced at a polynomial, because otherwise, it cuts itself. There are some notable exceptions, as any plane curve, where the function P( ) = 0 generates the plane of the respective curve as a surface of least area. (4) The aspect of some surfaces, as those which we mentioned before, could be studied, based on a package of programs which make numerical calculus, plotting of the cuts in the plane of the coordinates (0, v), construction, plotting and rotation of the generated minimal surfaces in R3. (5) All the constructed minimal surfaces are solutions of Plateau’s problem in the sense of [l], the required continuous zone, without singular points being at least the reunion of some neighborhoods, by one around every point of the generating curve. (6) One sees that for the construction of a minimal surface as before, it is necessary and sufficient to exist the field of tangent vectors at the curve, denoted c, and the regular enough field of vectors along the curve, denoted a, such that a(@)-~ti(@) and la(@)1 = IC(O)l, V 0 E 0. If c and (I belong to L(O), then the existence of the minimal surface is assured. Generally speaking, we can define the field a, without using the Frenet’s frame of the curve. Therefore, the method can be used on curves for which the oneness of the Frenet’s frame is not assured. Thus, if c( ) give the parametric coordinates of a straight line-for example, choosing c,(O) = ~~(0) = 0, ~~(0) = Oand taking u,(O) = cos(cyO), u,(O) = sin(&), u,(O) = 0, CY# 0, one generates a right helicoid, which its central axis is the generating straight line.
8. Some indications on the figures On all the figures, the generating curve is green. In all cases, we drew coordinate curves Y = ct. Also, we needed some time to complete few figures with curves 0 = ct. We represented the continuous zone of the minimal surface generated by the curve c,(O) = cos(O), c*(O) = sin(@), ~~(0) = 0.3 cos(20) and by P(0) = 0. The parameter v is varying between -0.48 and +0.48. One observes that exactly before the appearance of cuts, for IV] increasing, the coordinate curves I, = ct from the external zone, converge at a kind of spatial square, in the same time, the internal coordinates curves tending to a spatial cross with four arms. Generally speaking, if cg( ) is a trigonometrical polynomial of ‘N’th degree, the margin of the internal continuous zone is a cross with 2N arms and the margin of the external one is a polygon with ‘2N’ sides. Also, at least in the case of a curve on a cylinder and for a tixed function P, one can prove that if the generating curve ‘tends’ to be a circle, then the minimal surface ‘tends’ to be sagging, the continuous zone will be greater and the central cross tends to be more and more little and closed. At the limit, when the curve is a circle, the limits of the continuous zone tend to be infinite and the cuts vanish. Fig. l(a) presents this zone, seen from the axis 02, and Fig. l(b) offers a lateral view of the surface. For the previous curve and function P, one represents the starting portions of two adjoining external petals, in few successive positions. The parameter v is varying between -0.48 and -1.3. One sees how one petal ‘climbs’, in the same time, the other petal ‘descends’. The limits of the continuous zone are drawn with red. The other two undrawn petals are opposite and symmetrical given the drawn petal. The represented petals are fully generated for YE (-CQ, -0.48) and for 0 E (7r/4,31~/4), respectively (37r/4,51T/4). For the same curve and function P, we drew the generating curve v = 0 and the margins of all the four external petals (V < 0), through curves 0 = ct. We see from the axis 02 in Fig. 3(a) and from a ‘lateral point’ in Fig. 3(b). Minimal surface passing through the circle c,(O) = cos(O), ~~(0) = sin(O), c,(O) = 0, generated by P(0) = 0.45 cos(20). The parameter v is varying in the interval [0, +0.8]. As we know, such a surface
V. Visarion, M. Baubec
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141 (1997) 311-333
(W
lb) F1g.1. Fig. 2.
(cl
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K Visarion, M. Baubec
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(a)
141 (1997) 311-333
(a)
(b) Fig. 3. Fig. 4.
Fig. 5. Fig. 6.
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is not cut. The surface is seen from a ‘lateral point’ in Fig. 4(a) and from the plane of the circle in Fig. 4(b). Catenoid passing through the same circle, generated by the constant function P(0) = n/4. We plotted the cuts in the plane of coordinates (0, V) - 0 E [0,21r] and Y E [0,4]-, for the minimal surface generated by the curve c,(O) = cos(@) c?(O) = sin(@) c3(0) = 0.1 cos(0) + 0.1 sin(@) + 1.2 cos(20)
+ 0.5 sin(20)
+ + 0.02 cos(30)
+ 0.1 sin(30)
The cuts are white, the axis 00, OV and the lines 0 = 21r, v = 4 are red. One sees that due to the continuous zone, the cuts ‘start’ from a certain distance of the axis 00 and they tend to become parallel with the axis 0~. The points which cannot be linked with the real axis 00 through a continuous line which do not cross the cuts, must be excluded from the domain of definition of the surface, because the respective domain must contain the axis 00 and it must be simply connected. The cuts were found, solving in a numerical way Eqs. (6.10).
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll]
L. Bianchi, Lezioni Di Geometria Diferenziale (Enrico Spoelli, Pisa, 1902). K. Weierstrass, Mathematische Werke, Vols. l-7 (Edit. Mayer Berlin, 189441927). R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York-London, 1953). J. Nitsche, A new uniqueness theorem for minimal surfaces, Arch. Rat. Mech. Anal. 52(4) (1973). F. Tomi, On the local uniqueness of the problem of least area, Arch. Rat. Mech. Anal. 52(4) (1973). J. Nitscwlesungen iiber Minimalplachen (Springer, Berlin, 1975). V. Visarion and C. Stanescu, Calculul Starilor de Tensiune in Teoria Placilor Curbe (Le calcul des Ctats de tension dans la theorie des coques. Cylindres, surfaces d’aire minime) (Edit. Academic Roumaine, Bucharest, 1969). G.M. Fihtenholtz, Course of Intergral and Differential Calculus (Edit. Tehnica, Bucharest, 1969). R. Osserman, A Survey of Minimal Surfaces, 2nd edition (Dover Puublication, New York, 1986). C. Costa, Uniqueness of minimal surfaces embedded in R’ with total curvature 12rr, J. Diff. Geom. 30(3) (1989). D. Hoffman, Comment utiliser un ordinateur pour trouver de nouvelles surfaces minimales et des bulles de savon, University of Massachusetts, MA 01003.