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4D Part 3: Minimal Surfaces
Posted:
May 27, 1993 10:49 AM
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Given a fixed boundary in nD, what n-1 dimensional
hypersurface with this boundary has the minimum area?
This is the question to answer in the field of minimal
surfaces, studied by Ken Brakke and John Sullivan of the
Geometry Center. In three dimensions, this question
becomes: given a boundary in 3D, such as a loop or shape
made out of wire, what is the surface with minimum area
with this wire as its boundary? The answer should be quite
familiar to everyone; the soap film formed by dipping a
wire frame in soap solution gives a surface of minimum
area with that boundary.
Sometimes these surfaces are unexpected. For example, a
tetrahedral frame makes a soap film with planes starting
at each edge, all meeting at a point in the center, whereas
a cubic frame makes a soap film with surfaces headed
towards the center but which meet at an interior square
with rounded edges. (Many science museums have soap
bubbles and frames for people to play. It is also not
difficult to make solution and wire frames.)
Although it is easy to obtain minimal surfaces, they are
mathematically difficult to describe. Even for the
simple case of a cubic boundary, there is no known
mathematical equation which describes the soap film.
Even after gaining an understanding of the mathematics,
researchers always are going back to soap films.
"Whenever I get stuck, I take out the wire frames and soap
solution again," says Brakke.
Sullivan looks at n-1 dimensional minimal
hypersurfaces in nD, where n is arbitrary. He observes
that there are hypersurfaces which are locally minimal
but not necessarily globally minimal; in other words,
two different surfaces spanning a given boundary may
have the property that each minimizes area with respect
to perturbations in a small neighborhood of each point of
the surface. For example, in 3D, given two parallel
circles as a boundary wire, one can get two different soap
films: namely, a flat disk inside each circle
separately, and a catenoid connecting the two circles.
The catenoid surface is the shape you get if you form a
surface of rotation using the St. Louis Arch, rotating
around a line above it and in its same plane.
Sullivan finds general conditions to determine which of
the locally minimal surfaces actually has the minimum
area. His conditions apply any arbitrary dimension, so
he does not actually have to work specifically with
problems in 4D.
Brakke looks at higher dimensional singularities in
minimal surfaces. Singularities are just the possible
ways in which a surface could be lacking in smoothness.
For example, in the tetrahedral boundary case, the
planes starting at the edges intersect three at a time
along lines toward the center. These lines are singular.
Also, the planes all meet at a singular point in the
center.
The two singularities above are both mathematical cones
on some geometric object, where the cone on an object is
defined as all the lines connecting that object to the
origin. The first singularity was three planes meeting
along a line at 120 degrees. This is the cone on the
vertices of a triangle extended into 3D. The second
singularity was six planes meeting at a point. This is the
cone on a tetrahedral wire frame.
Jean Taylor proved the two types of singularities
described are the only possibilities in three
dimensions. See the article on Taylor's work in the
May/June issue of The Sciences. Thus Brakke is trying to
classify singularities for minimal hypersurfaces in
higher dimensions.
Since each dimension builds on the singularities of the
previous dimensions, Brakke concentrates his work
specifically on 4D. He has been successful; using his
program Surface Evolver to help guess possible surfaces
with singularities, then checking his guesses
analytically, Brakke proved that the 3D cone on the 2D
hypercube frame in 4D is a minimal surface. In fact, in all
dimensions higher than three the n-1 dimensional cone on
the n-2 dimensional hypercube frame is a minimal surface
with a singularity.
The new singularity, along with singularities obtained
by extending 3D surfaces to 4D hypersurfaces, is almost
the whole story in the 4D case. Brian White proved that all
possible singularities occur on flat sided cones in 4D;
the result is specific to 4D and not true in 5D. Based on
White's result, Brakke was able to show that there were
only a finite number of cases of singularities in 4D minimal
surfaces; for all but the simplex frame, hypercube frame, and one other
case, he proved that these surfaces were not minimal.
Brakke is currently trying to prove or disprove this last
case; this would complete the 4D classification.
Brakke says that even after his work in 4D, he is unable to
really visualize it. His responded with skepticism to
the idea that anyone can visualize 4D. He said, "Give them
a test. Show them a bunch of projections of similar 4D
objects and ask them which are pictures of the same
object. Show them one 'side' of a 4D object, and have them
describe the other 'side.' I think you'll find that
nobody can really visualize 4D. The best they can do is try
to understand a few theorems and see that the objects
obeys these theorems."
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