CONTINUOUS MG-DEFORMATIONS OF SURFACES WITH BOUNDARY IN EUCLIDEAN SPACE
The properties of continuous deformations of surfaces with boundary in Euclidean 3-space preserving its Grassmannian image and product of the principal curvatures are studied in this article. We determine the continuous MG-deformation for simply connected oriented surface F with boundary дF in Euclidean 3-space. We derive the differential equations of G-deformations of surface F. We prove the set of lemmas where we derive auxiliary estimations on norms of functions characterizing MG-deformations of surface F. Then on the surface F we introduce conjugate isothermal coordinate system which simplifies the form of equations of G-deformations. From the system of differential equations characterizing G-deformations of surface F in conjugate isothermal coordinate system we go to the nonlinear integral equation and resolve it by the method of successive approximations. We derive the equations of MG-deformations of surface F. We get the formulas of change ∆(g) and ∆(b) of determinants g and b of matrixes of the first and the second fundamental forms of surface F, respectively, for deformation {F <sub>t</sub>}. Then, using formulas of ∆(g) and ∆(b), we find the conditions characterizing MG-deformations of two-dimensional surface F in Euclidean space E <sup>3</sup>. We show that finding of MG-deformations of surface F brings to the following boundary-value problem (A): d <sub>z¯</sub>ẇ + Аẇ + Bẇ¯ + E (ẇ) = Ψ <sup>∙</sup>, Re{λ¯ẇ} = φ <sup>∙</sup> on dF, where А, В, λ, Ψ <sup>∙</sup>, φ <sup>∙</sup> are given functions of complex variable, ẇ is unknown function of complex variable, operator E(ẇ) has implicit form. Prior to resolving boundary-value problem (A) we find the solution of the following boundary-value problem for generalized analytic functions: d <sub>z¯</sub>ẇ + Аẇ + Bẇ¯ + E (ẇ) = Ψ <sup>∙</sup>, Re{λ¯ẇ} = φ <sup>∙</sup> on dF, Then we use the theory of Fredholm operator of index zero and the theory of Volterra operator equation. Using the method of successive approximations and the principle of contractive mapping, we obtain solution of boundary-value problem (A) and the proof of theorem 1, the main result of this article.