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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
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On the Problem of Plateau (Paperback, 1993 ed.)
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On the Problem of Plateau (Paperback, 1993 ed.)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, 2
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The most immediate one-dimensional variation problem is certainly
the problem of determining an arc of curve, bounded by two given
and having a smallest possible length. The problem of deter points
mining and investigating a surface with given boundary and with a
smallest possible area might then be considered as the most
immediate two-dimensional variation problem. The classical work,
concerned with the latter problem, is summed up in a beautiful and
enthusiastic manner in DARBOUX'S Theorie generale des surfaces,
vol. I, and in the first volume of the collected papers of H. A.
SCHWARZ. The purpose of the present report is to give a picture of
the progress achieved in this problem during the period beginning
with the Thesis of LEBESGUE (1902). Our problem has always been
considered as the outstanding example for the application of
Analysis and Geometry to each other, and the recent work in the
problem will certainly strengthen this opinion. It seems, in
particular, that this recent work will be a source of inspiration
to the Analyst interested in Calculus of Variations and to the
Geometer interested in the theory of the area and in the theory of
the conformal maps of general surfaces. These aspects of the
subject will be especially emphasized in this report. The report
consists of six Chapters. The first three Chapters are important
tools or concerned with investigations which yielded either
important ideas for the proofs of the existence theorems reviewed
in the last three Chapters."
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