Constant mean curvature surfaces via an integrable dynamical system

Abstract

It is shown that the equation which describes constant mean curvature surfaces via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its simplest finite-dimensional reduction is the integrable Hamiltonian system with two degrees of freedom. This finite-dimensional system admits S 1 -action and classes of S 1 -equivalence of its trajectories are in one-to-one correspondence with different helicoidal constant mean curvature surfaces. Thus the interpretaion of well-known Delaunay and do Carmo-Dajzcer surfaces via integrable finite-dimensional Hamiltonian system is established.Surfaces, interfaces, fronts, and their dynamics are key ingredients in a number of interesting phenomena in physics. They are surface waves, growth of crystal, deformation of membranes, propagation of flame fronts, many problems of hydrodynamics connected with motion of boundaries between regions of differ densities and viscosities (see, e.g., [1,2]). Quantum field theory and statistical physics are the important customers of surfaces too (see [3,4]).Mean curvature plays special role among the characteristics of surfaces and their dynamics in several problems both in physics and mathematics (see, e.g., [5,6]). Surfaces of constant mean curvature have been studied intensively during last years (see, e.g., [7,8,9]).

Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions

“…This is very similar to the Weierstrass representation for weakly conformal immersions in R 3 due to B. G. Konopelchenko [Ko]. Some variants were proposed also in [Ke] and this representation has been studied by many authors [KoT1,KoT2], [B1, B2], [KuS], [T1, T2]. Actually we shall first present our representation using notations that make evident the similarities between the two theories, since they rely on a kind of Dirac equation.…”

SusyCPN−1model and surfaces in

“…The subject of Weierstrass representations of surfaces immersed in multidimensional spaces was introduced few years ago by Konopelchenko et al 1,2 . This has generated interest 3,4 in looking at the properties of these surfaces and relating them to the solutions This approach involved writing the equation for the harmonic map as a conservation law and then observing that in this construction a special operator played a key role.…”

Split-Quaternionic Representation of the Moving Frame for Timelike Surfaces in 3-Dimensional Minkowski Spacetime

“…Recently, the Gauss-Codazzi equations for surfaces in Euclidean three-space were established from a three by three matrix representation and a quaternionic representation was introduced for the moving frame of the conformally parametrized surface (Konopelchenko and Taimanov, 1996). This work is extended here in what is intended to be a companion paper (Konopelchenko and Taimanov, 1996;Bracken, 2004).…”