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Soap Films and Problems without Unique Solutions: Many complicated everyday phenomenacan be seen as geometry problems in higher-dimensional spaceAuthor(s): Frank MorganSource: American Scientist, Vol. 74, No. 3, Special Book Issue (May-June 1986), pp. 232-236Published by: Sigma Xi, The Scientific Research SocietyStable URL: http://www.jstor.org/stable/27854095 .

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Soap Films and Problems without Unique Solutions

Frank Morgan

Many complicated everyday phenomena can be seen as geometry problems in

higher-dimensional space

Soap films beautifully illustrate how apparently simple problems can have many different solutions. Of course, in general problems, the lack of a unique answer can

complicate immensely the numerical computation of solutions. A computer analysis of a proper business

strategy or of the best shape for an airplane wing could be slowed or even confounded by the existence of many different solutions, much as a person's actions are slowed if he has many close decisions to make. In either

case, the signposts along the way, whether mathemati cal or mental, are unclear and contradictory, and the resolution becomes more difficult.

It may at first seem paradoxical that a mathematical

problem can have many solutions, but it can. This article uses soap films to illustrate such nonuniqueness and to indicate some of the underlying general principles.

Consider the problem of detenrdning what soap film will form on a given wire boundary (2-3). Many boundaries support two or more different soap films (4, 5):

Dipping such wire boundaries in a soap solution sometimes yields one surface, sometimes the other,

according to how the wire is removed from the solution. For a very few boundaries, there are two different films with exactly the same area:

Frank Morgan is Cecil and Ida Green Career Development Professor of Mathematics at MIT. His work in the calculus of variations explores minimal

surfaces in all dimensions. He likes to use common soap films to suggest and

illustrate advanced geometric ideas. Address: Department of Mathematics,

MIT, Cambridge, MA 02139.

Soap films provide a concrete example of how a mathematical

problem can have more than one solution. Although such films

always tend to form a surface of minimum area, some wire frames

can support two or more different films. In this case one film

extends around the front of the frame, the other around the back.

(Photographs courtesy of Gordan Gahan/Prism; drawings 1-10,12

by Jim Bredt.)

232 American Scientist, Volume 74

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There are even boundaries like this one

which support many different soap films:

With two choices at each of the three locations, there are at least 2x2x2 or eight different soap films that can form on this boundary.

But can a boundary support infinitely many soap films? In 1950 the eminent mathematician Richard Cou rant (4, see also 5, 6) described an example whose main idea is conveyed by this frightening picture:

He argued that with two choices at infinitely many locations, there would theoretically be infinitely many soap films. However, Courant himself considered this

boundary so badly twisted at the right end as to be unfair.

Then can a nice, smooth boundary support infinite

ly many soap films? Even in this case, the answer turns out to be yes (7).

To prepare for the example that establishes this, first consider four parallel wires:

They bound two different soap films, each consisting of two strips of surface:

To fit them together requires a crosspiece

which could be smoothed to a saddle:

1986 May-June 233

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The way in which a soap film forms on a given frame can offer

insight into solutions to mathematical problems in higher dimensions. Here the author examines soap films supported by an

intricately interlaced wire boundary.

The example of a boundary B that can support

infinitely many soap films consists of four such wires

bent into four circles:

Two of the circles are concentric (green, red ), a third circle lies between and above them (blue), and a fourth circle lies between and below them (black).

One soap film with this boundary consists simply of two separate strips of surface, each bounded by two of

the circles:

But another soap film consists of strips which switch at two crosspiece saddles, placed at opposite points and

balancing each other. In fact, there can be any even

number of evenly spaced crosspiece saddles and holes in a soap film that looks something like this:

Furthermore, any one such soap film gives rise to

infinitely many others as it is rotated about a vertical axis. Such a rotation changes the locations of the cross

piece saddles and holes, but leaves the boundary looking the same.

Unfortunately, these wonderful soap films are un

stable and will probably never be observed in nature or

in the laboratory. The slightest disturbance would upset the balance; holes would collapse, until only two sepa rate strips of soap film, each bounded by two of the wire

circles, would remain. Soap films naturally tend toward

such states of least area. Can a smooth boundary support infinitely many

surfaces of least energy (least area)? By 1979, the work of

Tomi (8) and Hardt and Simon (9) showed the answer to

be no. Their theorem can be stated thus: A smooth

boundary curve in R3 supports only finitely many oriented

area-minimizing surfaces. The term ''area-rninimizing7' means that no other surface with the same boundary has

less area. The word "oriented" indicates that we consid er only ordinary two-sided surfaces and exclude one

sided nonorientable surfaces like the M?bius strip:

If nonorientable surfaces are allowed, the question re

mains open: it is unknown whether a smooth boundary curve can support infinitely many nonorientable area

rninimizing surfaces. So far we have stayed in R3, ordinary, undistorted

three-dimensional space. But science has come to think

of our physical universe as curved. It happens that in

many curved spaces it is possible for a smooth curve to

support infinitely many oriented area-minimizing sur

faces. A lower-dimensional analogy is provided by a

234 American Scientist, Volume 74

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round sphere, which has infinitely many shortest paths from the south pole to the north pole:

Notice how the whole sphere arises front taking any one

of these shortest paths and revolving it around the vertical axis. .

Let me d?sdribe on? example of infinitely many area-minimizing st?ifaces with a common, boundary, inside a curved three-dimensional space, or Riemannian

manifold> M. M itself will sit in ordinary four-dirhen sional space, R4. To build it, start with the following complicated, somewhat arbitrary two-dimensional sur face S in R3 : k

Notice that the boundary S consists of two circles at the bottom in the x-y plane.

. '

; In order to get many copies of S with ex?c?y th?

same boundary, first consider revolving S ?b?ut the y axis in RJ(i.e., rotating the z axis toward th? * axis).

Unfortunately, this does not quite work: although the y axis does not move, the boundary circles rotate. ,

In&te?d, add a new "w axis" in a fourth dimension, and r?tate the 2 axis toward the w axis arid all the way aro?nd in a full circle. Now the x axis arid th? y axis both remain fixed, and so does the whole x-y plane, including th? two boundary circles. The rest of S revolves in a big circle to fill out a curved three-dimensional manifold M in R4. The picture at the right shows two separate instants during the revolution ?f S (of course there is another copy of S for each instant of the revolution).

Although the bulging surface S is clearly not area minimizing in RJ, it is area-minimizing inside the new

curved universe M, and so are all of its copies as it revolves. These copies, one for ?ach instant of the r?volution process, provid? the desired example of infi

nitely many area-minimizing surfaces with the same

boundary.

Despite s?ch examples, I proved just a few years ago (10) that under certain hypoth?ses ? smooth bound ary in a complete, connected Riemannian manifold M

supports only finitely many are?-minirnizing surfaces. Here are the three hypotheses: (1) Unlike the above example, M must be noncompact, that is, infinite in extent. (2) M must be real-analytic, that is, varying so

smoothly from point to point that its precise behavior at one point d?termines its behavior everywhere ?ls?.

(3) There must be some limit |o how much M curves; Specifically, the sectional curv?ture must be bounded above, and the injectivity radius must be bounded away from 0. (The concept of curvature is first defined for two dim?nsional surfaces;, the sectional curvature of a three dimensional manifold is then Simply th? curvature ?f its twchdimehsi?nal slices or "sections." The concept, of an

injectivity radius depends on a phenomenon that hap pens ?rdy in curved space, tight beams emitted in different directions can meet up again at some distance. Th? shortest distance in which this can occur is called the injectivity radius, ?or example, on the earth, one can

imagine signals emitted from the south pole meeting up again at the north pole. In this cas?, the injectivity radius is simply the distance from one pole to th? other.)

Under these three hypotheses; a smooth boundary curve in M supports only finitely many ?rea-minimizing surfaces. The proof assumes that there are infinitely

many ?rea-minimizing surfaces arid, derives a contradic tion. Since the Surfaces are area-minimizing, they stay within some finite distance df the boundary, inside some region of finite size. Since there are infinitely many of them in that finite region, lots of them have to accumu

1986 May-June 235

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late at some point. Since M is real-analytic, it can be shown that they propagate from that point to fill the

whole manifold M. But since M is noncompact, this contradicts the constraint that they must stay inside some finite region.

What about higher-dimensional spaces? The dimen sion of a space is really just the number of parameters required to describe a location in the space. For example, the surface of the spherical earth is two-dimensional,

with location described by latitude and longitude. The whole physical universe is three-dimensional, with dis tance from the earth the usual third parameter. Adding time yields the four-dimensional universe of general relativity.

But there are other commonplace spaces to consid er, such as the space of all positions of your right arm

keeping your shoulder fixed. Each position of your arm is a single location in this space. It is described by the location of your elbow on an imaginary sphere centered at your shoulder (two parameters), the location of your wrist on an imaginary sphere centered at your elbow

(two more parameters), and the angle through which

your forearm is twisted or rotated (one more parameter). Hence the space of positions of your right arm is 2 + 2 + 1, or five-dimensional.

Thus mathematicians might tackle the engineering problems involved in designing robot arms as geometry problems in higher-dimensional spaces.

It turns out that our results ruling out infinitely many area-minimizing surfaces generalize to higher

dimensional spaces. For n ̂ 7, in ordinary n-dimension al space R", or in an n-dimensional manifold M satisfy ing the hypotheses above, a smooth (n

? 2)-dimensional

boundary supports only finitely many (n ?

^-dimension al oriented, area-minimizing surfaces. However, in and above, the question remains open. In high dimen sions singularities occur in area-minimizing surfaces and

make the problem much more difficult. It looks like seven-dimensional soap films in R^can have trouble some corners.

References 1. F. J. Almgren and J. E. Taylor. 1976. Geometry of soap films. Sei.

Am. 235:82-93.

2. C. V. Boys. 1958. Soap Bubbles. Dover.

3. Bruce Schecter. 1984. Bubbles that bend the mind. Science 84, March.

4. R. Courant. 1950. Dirichlet's Principle, Conformai Mapping, and Miminal Surfaces. Interscience.

5. Johannes C. C. Nitsche. 1969. Concerning the isolated character of solutions of Plateau's problem. Math. Z. 109:393-411.

6. Paul L?vy. 1947-48. Le probl?me de Plateau. Mathematica 23:1-45.

7. Frank Morgan. 1981. A smooth curve in R3 bounding a continuum of minimal manifolds. Arch. Rat. Mech. Anal. 75:193-97.

8. Friedrich Tomi. 1973. On the local uniqueness of the problem of least area. Arch. Rat. Mech. Anal. 52:312-18.

9. Robert Hardt and Leon Simon. 1979. Boundary regularity and embedded solutions for the oriented Plateau problem. Annals of

Math. 110:439-86.

10. Frank Morgan. 1983. On finiteness of the number of stable minimal

hypersurfaces with a fixed boundary. Bull. AMS 13:133.

"As you know, we're veggies."

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