Lorentzian varifolds and applications to closed relativistic strings

Abstract

We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian minimal surfaces, we introduce, within the varifold class, various notions of generalized minimal timelike submanifolds of arbitrary codimension in flat Minkowski spacetime, which are global in character and admit conserved quantities, such as relativistic energy and momentum. In particular, we show that stationary lorentzian 2-varifolds properly include the class of classical relativistic and subrelativistic strings. We also discuss several examples.