Fundamental Equations And Lax Pairs Of Weingarten Surfaces

In this thesis we study Weingarten surfaces in three-dimensional Euclidean space whose two principal curvatures satisfy a cubic function relationship. The thesis is organized as follows.In Chapter 1 we first review the classical Backlund transformation, a transformation on pseudo-spheres in three-dimensional Euclidean space. We also introduce two pseudo-spheres constructed by applying the classical Backlund transformation to some trivial surfaces; Then we introduce some generalizations of the classical Backlund transformation, such as Backlund transformation on surfaces for which the Gaussian and mean curvatures are linearly related; Finally we state the problem studied in this thesis.In Chapter 2 we introduce the surface theory from the point of view of integrable system. For a surface in three-dimensional Euclidean space, its Gauss-Codazzi equations are the compatibility conditions of its Gauss-Weingarten equations, therefore the Gauss-Weingarten equations can be regarded as the Lax pair of the Gauss-Codazzi equations. Moreover, since there exists a homomorphism from SO(3, R) into SL(2, C), the Lax-pair in 2 by 2 forms can be constructed. For Weingarten surfaces, by choosing the lines of curvature as its parametric curves, one can solve the Codazzi equations by integration.Chapter 3 is the main part of this thesis. In this chapter we study the integrability of Weingraten surfaces for which the principal curvatures satisfy a cubic function. We classify the Gauss-Codazzi equations for such surfaces, and at the same time we obtain their Lax pairs. According to the roots of a cubic equation, we divide our discussions into four cases. In each case, we solve the Codazzi explicitly, and then obtain the Gauss equation and its Lax pair.