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THE EXISTENCE OF SURFACE PATCHES FOR PERIODIC MINIMAL SURFACES
J. Nitsche
To cite this version:
J. Nitsche. THE EXISTENCE OF SURFACE PATCHES FOR PERIODIC MINIMAL SURFACES.
Journal de Physique Colloques, 1990, 51 (C7), pp.C7-265-C7-271. �10.1051/jphyscol:1990727�. �jpa-
00231126�
THE EXISTENCE OF SURFACE PATCHES FOR PERIODIC MINIMAL SURFACES
J. C . C. NITSCHE
University of Minnesota, U.S.A.
1
From the very beginning - 1865, when he had just turned twenty-two
-
Hermann Amandus Schwarz visualized his solution surfaces of Plateau's problem for specific skew quadrilaterals also as fundamental domains (surface patches) for the creation of triply periodic minimal surfaces embedded in R3 : already in [ l l ] , pp. 3-5, he describes clearly the labyrinthic nature with which these surfaces permeate space, dividing it into two highly intertwined regions of infinite connectivity. Of course, for him they were purely geometrical constructs ; there is no hint at all about their possible role as physical interfaces (not even Joseph-Antoine-Ferdinand Plateau who entertained scientific contacts with Schwarz had such notions) ; for this aspect, see the comments in [8], pp. xiii-xvi,5
279, A8.32. Subsequently, the associate fundamental domains, which meet the boundaries of certain tetrahedra a t right angles, yielded further examples of periodic minimal surfaces, also already studied by Schwarz. Moreover, in view of a general reflection principle, the latter are suitable as basis for the construction of triply periodic surfaces having constant, but non-vanishing, mean curvature H.' After numerous early contributions by E. R. Neovius, G. Tenius, A. Schoenflies, E. Stenius, 0 . Nicoletti, F. Marty, B. Steflmann, M. Wernick, T. Nagano, B. Smyth and others2, all, with only few exceptions, being restricted t o the discussion of simply connected fundamental domains bounded by four (straight or curves) geodesics, far more elaborate fundamental domains have recently been introduced, notably by A. H. Schoen [l21 and by W. Fischer & E. Koch [4]. They lead to an astounding wealth of new periodic minimal surfaces of distinct crystallographic types.The mathematical existence of these domains is as yet unsecured in most cases and thus poses interesting questions. Note e. g. the intriguing paraphrase in W. Fischer & E. Koch [4], p. 168 : "The existence of minimal surface patches with the shape of branched catenoids has been proved by soap-film experiments...". For example, the 'branched catenoids' and 'multiple catenoids' of Fischer-Koch are bounded by two closed curves lying in parallel planes. One, or both, of these curves have self-intersectionss, a fact which rules out the existence proof on the basis of J. Douglas's method [3] or the version developed by R. Courant in [2], chapter IV : it is an essential element of the classical approach t o the existence problem for minimal surfaces of higher topological type that the bounding contours are required t o be Jordan, z. e . simple closed, curves.
1 An existence proof for a certain interval
I
HI
< H o can be found in the author's lecture [g] ; see further [ l ] and [5].2 For references see e. g. [7],
3
818.3 A special example, foliated by its level lines, can be seen i n figure 72 on p. 512 of [7].
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990727
COLLOQUE DE PHYSIQUE
2
In the present note, it will be indicated how the above mentioned methods can be modified and extended so as to fashion an existence proof for the case of self-intersecting boundaries, as long as the aspect (the ratio of the curve diameters and the distance of the planes) remains in a certain range.
An analysis of the standard existence proofs shows that the simple closed character of the boundary components is imperative in two of the steps : (i) the presence of an embeddedness function (see [8],
5
23) which, as R. Courant and L. Tonelli have shown, allows the demonstration of the compactness of cohesive minimizing sequences with the help of Lebesgue's lemma, as well as the validity of Douglas's sufficient condition ; (ii) the proof that stationary vectors of Dirichlet's integral satisfy the relation %(W) = 0, i. e. are minimal surfaces, requires the possibility of unrestricted variations of the independent variables.The operative assumption which makes the new proof work is the observation that the bounding contours proposed by the crystallographers possess certain symmetries, despite their self-intersections. Correspondingly, the solution surfaces sought will be 'angesetzt' with the same symmetries ; furthermore, the embeddedness function will be suitably partitioned. As will be seen, this procedure compensates also effectively for the loss of freedom in the variation of independent variables.
3
Because of space limitations, the description to follow will concentrate on one specific, albeit representative, case4, that of a twofold branched catenoid shown on p. 167 of [4]. The bounding contours depicted here in Figure 1 consist of a square I'l of side 1 (to fix a scale) in the plane z = 0 and a figure eight (polygonal lemniscate) P2 composed of two squares of side $212 in the plane z = h, where the distance h
>
0 lies in a suitable range. Also, to compress an otherwise long exposition, reference will be made, whenever feasible, to$8
566-73 of [7] where the details of the classical existence proof for the case that and P2 are Jordan curves is expounded in detail. In fact, a familiarity on the reader's part with this proof, which is quite intricate belonging, as it does, t o the greatest mathematical achievements of the early twentieth century, associated with the names J . Douglas, R. Courant, C. B. Morrey, Jr. a. o., would seem to be quite desirable.4
I t will be our aim to find the minimal surface bounded by PI and
r2
as a surface in parametric representation S = {X = X(U,U) ; (u,w) ER),
where the position vector x(u,w) = (x(u,v), y(u,v), z(u,w)) is defined over the closure of a suitable ring domain R G R = {U, W, 0<
~2X
<
u2+
v2<
1). The inner radius r = rx of R will depend on x(z~,w). For convenience, we shall also4 Our method is general, however. The detailed eneral proof, along with selected other topics, was presented by the author in a graduate course fMath. 8310, Topics in Geometry) to a group of advanced studients at the University of Minnesota during the spring quarter of 1990.
combine the parameters U, v in the complex-valued form w = U
+
iv. To be admissible for our discussion (5 557), X is required to satisfy the following conditions :1) x(u,v) belongs to the regularity class M 0 fl Hi ; for a description of this function class see [8],
$5
193-227.2) x(u, v) maps the circles
l
Wl
= r andl
Wl
= 1 onto I' l and l?z respectively, in a monotone way. (Concerning the choice of orientation, see the remarks below).Denote by DR[x] =
!lR
(X:+
xi) du dv the Dirichlet integral and by d = d(rl, r2) the infimum of the values of DR[x] for all admissible vectors. Our disk-type minimal surface will now be the solution, if it exists, of the following variational problem(S
557) :To find an admissible vector X, with admissible parameter domain (i. e. 0
5
r = rx<
l), for which DR[x] = d(rl, I'z).The successful approach to this variational problem hinges on the existence of cohesive minimizing sequences
(5
558), e. g. minimizing sequences of surfaces which do not tend to degeneration. For sufficiently small positive values of the distance h, such sequences will exist($5
559, 565-6), provided the orientation of mappings from the boundary components
I wl
= r and1
wl = 1 of the parameter domain R to the curves l?l and r 2 , respectively, are chosen in a coherent way, to lead to an orientable surface, as the comparison surface in connection with Douglas's condition constructed in section 6 below. For the proof of this fact, as well as for the proof of the next step-
compactness of such minimizing sequences - t h e lack of an embeddedness function forCOLLOQUE DE PHYSIQUE
is detrimental. We can, however, divide into two parts, one part
rat)
lying in the half space y 2 0, the other part ra2), mirror image of the first, lying in the half space y5
0. The r i g , i = l, 2, are open arcs, but each has an individual embeddedness function T/')(E) defined as follows : Given E E (0, $2/2), consider any two points, yl, y2 on r i i ) at distance / y i-
y21 E. Let 6(yl, y2) be the diameter of the connected subarc of r!ji) with end points y l and y2. Then is the supremum of 6(yi, yz), taken over all positions of the yl, y2. We have T/~)(E) ) E, but for the case of the special curve r2 under discussion here, it is obvious that, in fact, T/')(E) = E.The partitioning of the contour I'2 into two subarcs necessitates a modification of the customary compactness proof for cohesive minimizing sequences, a modification which requires that admissible surfaces be further restricted by the following conditions of symmetry. These conditions reflect the symmetries of the bounding contours :
3) The components of the position vector x(u, v) satisfy the relations
4%
-v) = g u , v), y(u, -v) =-
y(u, W), z(u,-v) = z(u, v) as well as4-U,
v) = - 4 u , v), ),(-U, v) = y(u, v), z(-U, v) = z(u, v).It should be noted that the second set of conditions in 3 is appropriate for the case of a twofold branched catenoid considered here, but that its imposition is really optional.
With the modifications indicated, the existence proof for a solution of our basic variational problem can now be concluded. The position vector x(u, v) of the solution surface has the symmetries stipulated in condition 3 above. Moreover, X is a harmonic vector. As a consequence of this, the complex-valued expression @(W) = (X;
-
xi) - 2 k X is seen to be an analytic functionU v
of the complex variable W, expressible as a Laurent series @(W) =
lm
c% wn with complex?%--m
coefficients cn = an
+
ib,, convergent for rx< I
wl<
1, in fact, for rx5 l
wl < l , with the exception of eight points on the boundary dRx of Rx (the points which correspond to the corners of the bounding contours).5
While the stipulation 3 makes the variational problem tractable, the actual test will be our ability to show that x(u, v) is the position vector of a minimal surface, i. e. that the relation + ( W ) = 0 is satisfied. Customarily, this is done with the help of a variation of the independent variables
denotes the inverse of the mapping introduced. Then, if 12 is admissible, we find that :
Here the first variation V~[X;(P], after a somewhat lengthy manipulation, takes the form
Since X minimizes Dirichlet's integral, we must have Vl[x;~] = 0. The freedom in the choice of the function (P leads to the conclusion that @J must have the form @(U) = a-2 W-2. A second similar transformation (5 562) shows that a-2 must also vanish.
It should be noted that the two transformations of the independent variables alluded to here can be combined ; but this fact is of secondary importance at the moment.
The point to be kept in mind here is that the variations available to us, leading from X to 12, must not destroy the symmetries possessed by the solution vector. This means that the variation cp is subject to the conditions ( ~ ( u , v) =
-
(P(-U, v) =-
(P(%, -v). They imply the relations a(u,V) =
-
&(-U, v) = a(%, -v) and @(U, v) = B(-%, v) = - P(%, -v) on which the admissibility of 12rests. With this restriction, the conclusion @(W) = 0 is not in general justified, unless the function has suitable symmetries itself.
Fortunately, an inspection of the implications of condition 3 above reveals that @ satisfies the relations :
Here
w
= u-
iv. Thus it is seen that @ is available in an expansion of the special formm
=lm
a2n $ n with real coefficients a2n. A detailed discussion shows that under these n=--a,circumstances the relation @ ( U ) = 0 is valid, even for the restricted class of variations.
The proof of the existence of a "branched catenoid", i. e. a disk-type minimal surface bounded by the curves
r l
and r2 is now complete.6
Our minimal surface will exist as long as the distance between the planes of the curves l?l and
r 2
lies in a certain interval 0
<
h5
ho. The precise value of the existence limit h. is elusive. But the following can be said :COLLOQUE DE PHYSIQUE
It is a fact that the solution surface of our variational problem does not only minimize Dirichlet's integral, but also the area functional (in the class of admissible surfaces). As a consequence, Douglas's sufficient condition which, in turn, implies the existence of cohesive minimizing sequences, will be satisfied as long as ring-type comparison surfaces bounded by
rl
and
r2
can be found whose area is less that 2 (the total area of the squares spanned into the boundary). A ruled surface with generators connecting l?l and I'z, composed of eight congruent pieces of the part shown in Figure 1, left, is a simple candidate. Its area comes toA computation shows that this area is smaller than 2 for 0
5
h5
0.36063.On the other hand, general inclusion theorems imply that no solution can exist if h is too large ; see e. g. [6], 171, VI.3.1, [10]. A crude application shows that h can certainly not be larger than (1
+
1/2/2) sinhp = 1.13137. Here, p is the root of the transcendental equation p = cothp.Thus the existence limit h. is a number in the interval 0.36
5
h.5
1.13. The bounds for h.are by necessity unsharp, since Douglas's condition guarantees the existence of an absolutely area minimizing surfaces, but excludes unstable surfaces which are generally present. In the standard example, a catenoid spanned into coaxial unit circles in parallel planes at distance h, the existence limit is h. = 1.325 while Douglas's condition is satisfied for 0
5
h<
1.055.7
The solution surface is everywhere regular, i. e. a differential geometric surface, even in the corners of the bounding contours PI and
Pz,
that is, the components of the position vector X are everywhere real analytic functions of suitable local parameters. This is so, because the angles of the corners are 90". The same would be true if these angles were 60°, or 45", or 3 0 h t c . As functions of the parameters (U, v), these components have simple algebraic singularities in the points on dR corresponding to the corners of Fl and I'2.For information concerning the general behavior of a minimal surface near corners of its boundary, see
$5
357-60 of [8].References
[l] Anderson, D. M., Davis H. T., Nitsche H. C. C., & Scriven L. E. : Periodic surfaces of prescribed mean curvature. Advances in Chem. Phys. 77, Interscience, New York, 1990, pp. 337-96
106430.
[4] Fischer, W., & Koch E. : New surface patches for minimal balance surfaces. I. Branched catenoids, Acta Cryst. A 45 (1989), 166-9, 11. Multiple catenoids, Acta Cryst. A 45 (1989), 169-74, 111. Infinite strips, Acta C7yst. A 45 (1989), 485-90. IV. Catenoids with spout-like attachments, Acta Cryst. A 45 (1989), 558-63.
[5] Karcher, H. : The triply periodic minimal surfaces aof Alan Schoen and their constant mean curvature companions. Manuscr. math. 64 (1989), 291437.
[6] Nitsche, J. C. C. : Ein Einschliefiungssatz fiir Minimalflachen. Math. Ann. 165 (1966), 71-5.
[7] Nitsche, J. C. C. : Vorlesungen uber MinimalJlachen. Springer Verlag, Berlin-Heidelberg- New-York, 1975.
[8] Nitsche, 3. C. C. : Lectures on minimal surfaces, Volume I, Cambridge University Press, Cambridge-New-York-New ~ochelle-Melbourne-~ydney, 1989
[g] Nitsche, J. C. C. : Mathematik in Berlin, Born Konkreter Geometrie iiber die Jahrhunderte. In Wissenschaft und Stadt, ed. D . Heckelmann & 0 . Busch, Colloquium Verlag, Berlin, to appear.
[l01 Osserman, R., & Schiffer M. : Doubly-connected minimal surfaces. Arch. Rat. Mech.
Anal. 58 (1975), 285-307
[l11 Schwarz, H. A. : Gesammelte Mathematzsche Abhandlungen. Vol. 1, Springer, Berlin, 1890.
[l21 Schoen, A. H. : Infinite periodic minimal surfaces without self- intersections.
NASA Techn. Rep. D-5541, 1970
