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Books > Science & Mathematics > Mathematics > Mathematical foundations > Mathematical logic
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Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Paperback, 2001 ed.)
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Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Paperback, 2001 ed.)
Series: Lectures in Mathematics. ETH Zurich
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One of the most striking development of the last decades in the
study of minimal surfaces, constant mean surfaces and harmonic maps
is the discovery that many classical problems in differential
geometry - including these examples - are actually integrable
systems. This theory grew up mainly after the important discovery
of the properties of the Korteweg-de Vries equation in the sixties.
After C. Gardner, J. Greene, M. Kruskal et R. Miura 44] showed that
this equation could be solved using the inverse scattering method
and P. Lax 62] reinterpreted this method by his famous equation,
many other deep observations have been made during the seventies,
mainly by the Russian and the Japanese schools. In particular this
theory was shown to be strongly connected with methods from
algebraic geom etry (S. Novikov, V. B. Matveev, LM. Krichever. . .
), loop techniques (M. Adler, B. Kostant, W. W. Symes, M. J.
Ablowitz . . . ) and Grassmannian manifolds in Hilbert spaces (M.
Sato . . . ). Approximatively during the same period, the twist or
theory of R. Penrose, built independentely, was applied
successfully by R. Penrose and R. S. Ward for constructing
self-dual Yang-Mills connections and four-dimensional self-dual
manifolds using complex geometry methods. Then in the eighties it
became clear that all these methods share the same roots and that
other instances of integrable systems should exist, in particular
in differential ge ometry. This led K."
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