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Surfaces of revolution in terms of solitons. (English) Zbl 0896.53007

In [Transl., Ser. 2, Am. Math. Soc. 179(33), 133-151 (1997; Zbl 0896.53006), see the preceding review], I. A. Taimanov considered the general case of modified Novikov-Veselov deformations. In the present paper, the global Weierstrass representation and its spectral properties are studied. The main result here states that for any \(u\geq 1\) the \(n\)th modified Korteweg-de Vries (mKdV) deformation transforms tori of revolution into tori of revolution, preserving their conformal types. As a consequence, the unit sphere and the round tori are stationary points of mKdV deformations. Two problems on invariance of first integrals of mKdV flows are discussed.
Reviewer: C.-L.Tiba (Iaşi)

MSC:

53A05 Surfaces in Euclidean and related spaces
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 0896.53006
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