Generalized Bernstein property and gravitational strings in Born–Infeld theory

Abstract

Motivated by Born–Infeld geometric electromagnetic theory, we consider a series of nonlinear equations which extend the minimal surface equations and the related, generalized, Bernstein problems. We study the relation of these equations and the conditions which lead to the triviality of the solutions. We also study a non-Abelian extension of these equations and establish a gap theorem for the Yang–Mills–Born–Infeld fields. We then couple the Born–Infeld electromagnetism with a Higgs scalar field and obtain an existence theorem for the self-dual multiple cosmic string solutions on a closed surface characterized jointly by the first Chern class and the Thom class formulated over the hosting complex line bundle.