Requisite control of minimal surfaces out to infinity
was developed based on Osserman's work on the Weierstraß representation.
Out of this work one succeeded in a very clear way in changing even
already known surfaces, making them more complicated, even though this
change has an easily visible effect on only a small part of the surface.
This success appears to contradict the above described ,inheritability'
property of minimal surfaces, which usually react in the form of
catastrophic perturbations at a greater distance from a place where small
changes are made. Mathematicians did not expect an abundance of new
examples because of this effect.
Luquesio P. Jorge of Ceara Federal University (Brazil)
and William M. Meeks III of the University of Massachusetts at Amherst
(USA), were the first who-starting from the imagined shape of the surface
to be newly constructed-undertook the controlled alteration of a minimal
surface out to infinity. Out of the two funnel-shaped ends into which the
catenoid spreads out, they made three ends. From this approximate
representation, along with Osserman' s theory, they concluded in 1980 with
the Weierstraß representation formula for a new minimal surface: the
trinoid, in which three ends are fitted together. Today we can build new
ends at arbitrary places on the catenoid. The bidenoid in Fig
12 has two additional tiny ends on the lower portion of the catenoid.
These new complete minimal surfaces are, however, not
embedded, since they intersect themselves, as one can see as soon as one
observes a larger portion of the minimal surface. The ends of the trinoid
intersect each other. Also the little ends of the bidenoid cut the
remaining surface.
In the case of embedded minimal surfaces, on the one
hand, since long ago periodic surfaces like the helicoid as well as the
Scherk surface and the surfaces of Schwarz were known; but on the other
hand, only two non-periodic surfaces-the plane and the catenoid-were
known. It was expected that these two were indeed the only two such
surfaces. In any case, nobody was able to suggest what geometrical form a
third candidate surface might have - until the famous Costa-Hoffman-Meeks
surface was discovered (Fig 3).
