On functional of Gauss curvature

Abstract

The authors study the problem of constructing a Gauss curvature functional whose variation on admissible surface is determined by its gauss curvature. In their previous papers the functional of such a type had been found for axisymmetrical surfaces. The crucial point in its constructing was correspondence between differential properties of the function determining it and and variation of the first quadratic form of the surface. Besides it was very important that axisymmetrical surfaces admit global half-geodesic parameterization. Thus, in order to obtain functional yielding gausss curvature in the general case it was necessary to revise differential equation whose solution gave Gauss curvature functoional in axisymmetrical case. This problemm was solved by analytic continution of the solution into upper half-plane. Further the problem of global half-geodesic parameterization arises. It is yet unsolved problem in the general setting. But the authors earlier showed that continuously differentiable surfaces with positively determined first quadratic form admit almost global half geodesic parameterization, i.e. it is possible to find a familly of geodesic lines covering it up to the null Hausdorff measure. The authors following the general lines study of axisysymmetrical case deduce integral-differential equations whose solution give desirable variation of the first quadratic form . This variation in accordance with differential properties of the solution of differentiable equation for the function determining gauss functional solve the problem of gauss curvature functional for the surfaces lacking axial symmetry.