Gaussian curvature functional in the class of convex Liouville surfaces with boundary

Abstract

In the paper, a class of regular convex and bordered Liouville surfaces is considered. We deduce the non-liner Beltrami equation whose solutions permit to transform arbitrary isothermal pameterization into semi-geodesic one. Using the well-known representations of geodesic lines of Liouville surfaces, we prove that Beltrami equation turns to be linear one. Using geodesic lines of the surface, we construct also topological mapping on the set defined by the distribution of geodesic lines. We prove that this mapping is a solution of the Beltrami equation realizing passing from isothermal parameterization to the semi-geodesic one. Applying the theorem of solvability of Dirichlet problem for Monge-Ampere equation, we prove that the admissible surfaces admit non-trivial variations leading to admissible ones. As in the case of axisymmetrical surfaces we define functional of Gauss curvature on the class of admissible surfaces and prove that its first variation for some special variations of the admissible surface is determined by its Gauss curvature.