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Plateau Problem

Fig 2. Stable soap film spanned by
a wire. After a slight disturbance it jumps back to its original
position.
Video (1.9MB, 0.5MB)

The exact description of minimal surfaces has been the
aim of two centuries of mathematical research up to our time. In recent
years this field has enjoyed a lively renaissance with the help of modern
visualization techniques. To find an optimum (such as to find a minimal
surface) belongs to the central problems in many areas of life. Every day
we are trying to find the shortest distance or finish projects in the
least amount of time. In 1760 the versatile mathematician JosephLouis de
Lagrange (17361813), Professor at Turin, Berlin, and Paris, recognized
that the problem of area minimizing is a characteristic model problem for
many practical phenomena and that a detailed study of its properties would
lead to farreaching insight into many other socalled variational
problems. He opened up the mathematical investigation of minimal surfaces
leading later to the Plateau Problem:
Does there exist for every arbitrarily complicated boundary curve a
surface of least area?
In spite of the fact that soap films seem to easily
solve this question in the affirmative, for a long time all attempts at a
mathematical treatment were disappointing. Lagrange himself indeed
discovered the necessary conditions  socalled partial differential
equations  that a minimal surface must satisfy at each of its points, but
using the theoretical methods of that time, it was quite hopeless to try
to answer his question in general. The degree of complexity of the
occurring problems outreached the capabilities of the mathematical
craftman's tools of those days and it urged mathematicians of the next
generations to develop manifold techniques in different areas of
mathematics. Among those were differential geometry, complex analysis,
theory of partial differential equations and calculus of variations, to
name the major fields contributing to minimal surface theory.
