A geometric interpretation of the spectral parameter for surfaces of constant mean curvature

Considering the kinematics of the moving frame associated with a constant mean curvature surface immersed in S we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S. The application of the Sym formula to this linear problem yields constant mean curvature surfaces in E. Independently, we show that the Sym formula itself can be derived by an appropriate limiting process R→∞. Integrable nonlinear equations in 1 + 1 dimensions are distinguished by the existence of the linear problem or spectral problem, i.e., an associated system of linear equations, containing the so called spectral parameter (see, for instance, [16]). The integrability conditions for the linear problem are equivalent to the considered nonlinear system. Integrable systems played an important role in the classical differential geometry [12], and are more and more important in the modern differential geometry [5, 20, 21]. Some integrable systems are of geometric origin [3, 13, 18, 19]. Given a spectral problem we can construct a local immersion by the so called Sym formula [6, 19]. For instance, starting from the spectral problem for the sine-Gordon equation we get pseudospherical surfaces. The Sym approach gives probably the best correspondence between the geometry and spectral problems [7, 19]. The spectral problem is necessary for the application of various methods of the soliton theory, like the inverse scattering method, the Darboux-Bäcklund transformation or algebro-geometric solutions in terms of Riemann theta functions. The Sym formula allows one to use all these methods in differential geometry. In the differential geometry of immersed submanifolds we have always a typical pair: the linear system of Gauss-Weingarten equations and their compatibility conditions, the nonlinear system of Gauss-Codazzi-Ricci equations. To obtain a linear problem of the soliton theory we need to insert a spectral parameter into the Gauss-Weingarten equations under consideration (for more details and references see, for instance, [6]). In this paper we consider surfaces of constant mean curvature H 6= 0. Constant mean curvature surfaces immersed in 3-dimensional Euclidean space E3 appear in the problem of soap bubbles if the (constant) outer pressure on both sides of the bubble surface is Copyright c © 2006 by J L Cieśliński