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Modified Novikov-Veselov equation and differential geometry of surfaces. (English) Zbl 0896.53006

Buchstaber, V. M. (ed.) et al., Solitons, geometry, and topology: on the crossroad. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 179(33), 133-151 (1997).
The author gives short introductions to the (modified) Novikov-Veselov equation and to the generalized Weierstrass representation of surfaces in Euclidean space: the (modified) Novikov-Veselov equation is a \((2+1)\)-dimensional generalization of the (modified) KdV-equation which is integrable by the inverse scattering method. The generalized Weierstrass representation provides any conformal (analytic) immersion (of a Riemann surface) into Euclidean 3-space from a solution of the Dirac equation \(\partial_z\psi_1= U\psi_2\), \(\partial_{\overline z}\psi_2= -U\psi_1\) with potential \(U: \mathbb{C}\to\mathbb{R}\). The classical Weierstrass representation of minimal surfaces is obtained when \(U= 0\).
If the potential \(U= U(z,\overline z,t)\) also depends on a parameter \(t\) and satisfies the modified Novikov-Veselov equation, the generalized Weierstrass representation provides a deformation for the corresponding surfaces, the “Novikov-Veselov deformation”. Periodicity conditions are discussed, i.e., when a deformation for closed surfaces is obtained. In the case of the Novikov-Veselov deformation of tori, the conformal structure and the Willmore functional are proven to be invariant (Theorems 1 and 2). This provides a new ansatz for proving the Willmore conjecture, \(\int_{T^2} H^2dA\geq 2\pi^2\) for any torus \(T^2\subset \mathbb{R}^3\).
In the reviewer’s opinion this is a very interesting and well-written paper.
For the entire collection see [Zbl 0871.00030].

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.)
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