FEDOROV UNIVERSAL EQUATIONS IN THE STRING AND MEMBRANE THEORIES

Abstract

It is shown that the equation of minimal hypersurface in the Euclidean (or pseudo-Euclidean) space can be written as the universal Fedorov matrix equation with first-order partial derivatives. Time-like minimal surfaces in the pseudo-Euclidean Minkowski space describe the free motion of relativistic strings and membranes, whereas space-like minimal surfaces describe the potential in the nonlinear Born electrostatic. All of them are imaginary images of minimal surface of the Euclidean space. Spherically symmetric surfaces are found to be all the three types, the hypercatenoid of any dimensionality and its imaginary images. The Fedorov equations provide rich information on the minimal surfaces.