The most amazing improvement of the final many years within the research of minimum surfaces, consistent suggest surfaces and harmonic maps is the invention that many classical difficulties in differential geometry - together with those examples - are literally integrable structures. This concept grew up ordinarily after the $64000 discovery of the houses of the Korteweg-de Vries equation within the sixties. After C. Gardner, J. Greene, M. Kruskal et R. Miura  confirmed that this equation will be solved utilizing the inverse scattering process and P. Lax  reinterpreted this system by means of his recognized equation, many different deep observations were made throughout the seventies, generally by way of the Russian and the japanese colleges. particularly this thought was once proven to be strongly hooked up with tools from algebraic geom etry (S. Novikov, V. B. Matveev, LM. Krichever. . . ), loop thoughts (M. Adler, B. Kostant, W. W. Symes, M. J. Ablowitz . . . ) and Grassmannian manifolds in Hilbert areas (M. Sato . . . ). Approximatively in the course of the related interval, the twist or idea of R. Penrose, equipped independentely, was once utilized effectively by means of R. Penrose and R. S. Ward for developing self-dual Yang-Mills connections and 4-dimensional self-dual manifolds utilizing complicated geometry tools. Then within the eighties it grew to become transparent that every one those equipment percentage an identical roots and that different cases of integrable platforms should still exist, specifically in differential ge ometry. This led ok.
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