On Almost Global Half-Geodesic Parameterization

The problem of existence of Global Half-Geodesic Surface Parameterization is considered. The problem is well known and it is yet unsolved in general case. It is known that for the twice-differentiable surfaces it has local solution. At the same time example of paraboloid of revolution proves that it is not possible in the general case to use local nets in order to construct the global halfgeodesic ones. In order to solve the problem the authors follow the way leading to the construction of isothermal parameterization for the surfaces with positive first quadratic form. To this end they deduce partial differential equation for the mappings giving necessary parameterization. In the contrast with the case of isothermal parameterization when the equation is Beltrami equation corresponding to the homogeneous elliptic system this equation is essentially non-linear one. Besides the new system admits degeneration at the points where the Jacobian of the solution is equal to zero or infinity. The speed of degeneration strongly affecting properties of the solutions is also unknown. In order to surpass these difficulties the authors change the challenge. Instead of the geodesics covering the whole surface they propose to find the geodesics covering the surface up to the set of Hausdorff null measure. Using the theory of -quasiconformal mappings they construct nonregular generalized solutions of non-linear Beltrami equation that nevertheless detect the necessary family of the geodesics. The constructed theory permits to study non-classical equilibrium forms of liquid drops.

first quadratic form of the parametric surface, quasiconfor- mal mappings, generalized solution, non-linear Beltrami equation, Sobolev spaces, imbedding theorems, weak convergence of the functions, almost global half-geodesic parameterization