BONNET SURFACES AND PAINLEVE EQUATIONS

Introduction Let F be a surface in Euclidean 3-space without umbilic points. This paper studies the following Problem : To classify non-trivial one-parameter families F , 2 (?;) of isometries of F = F 0 preserving both principal curvatures. Since the Gaussian curvature is preserved by isometries one can reformulate the problem replacing "both principle curvatures" by "the mean curvature function". Let us specify what do we mean by a non-trivial family. We consider families of surfaces which do not diier by rigid motions. We suppose also that the surfaces and isometries are suucient smooth. The case of surfaces with constant mean curvature (CMC-surfaces), which all possess non-trivial isometries, is also excluded from our consideration. We suppose that the mean curvature is a non-trivial function on F. It turns out that the condition of possessing a one-parameter family F of isometries, preserving H, implies restrictive conditions on F. Moreover, all the family F can be described (see section 2) as a reparametrization of F itself. The problem is reduced to the problem of classiication of surfaces F. Since the problem formulated at the beginning of this introduction was rst studied by Bonnet, we call these surfaces Bonnet surfaces. The problem is classical and many mathematicians contributed to its solution. O. Bonnet himself showed in Bo] that besides the CMC surfaces there is a class of surfaces, depending on nitely many parameters, which allows non-trivial isometries preserving H. These results were developed further by L. Raay, who proved that the Bonnet surfaces are isothermic (i.e. allow conformal curvature line parametrization) and isometric to surfaces of revolution. J.N. Hazzidakis H] showed that the mean curvature function H satiies an ordinary diierential equation of the third order and was able to integrate it once. Graustein G] proved that all Bonnet surfaces are Weingarten surfaces, i.e. the 1 mean and the Gaussian curvature are related d H ^ d K = 0. He also found a convenient alternative description for the Bonnet surfaces. Namely, he showed that these surfaces can be characterized as isothermic surfaces, where the function 1=Q with Q = 1 4 < F xx ? F yy ; N > is harmonic, that means (@ xx + @ yy)1=Q = 0: In modern notations Q is the Hopf diierential, written in isothermic coordinates x, y. Later the problem was treated by E. Cartan in C], where the most detailed results concerning …