Existence Results for the Nonlinear Hodge Minimal Surface Energy

Abstract

Given a compact Riemannian manifold $(M^n,g)$ and a fixed cohomology class, $[{\alpha }^{\ast }]\in H^k(M)$, we consider the existence of a minimizer $\alpha \in [{\alpha }^{\ast }]$ of the generalized minimal surface energy ${\int }_M\sqrt{1+|\alpha {|}^2}d{V}_g$. When $k=1$, we prove the existence of unique minimizers for every cohomology class $[{\alpha }^{\ast }]$. Next, when $k>1$, we construct examples of singular solutions for finite cohomology class $[{\alpha }^{\ast }]\in H^k(S^k\times S^k,g)$, where $g$ is conformal to the standard metric on $S^k\times S^k$. Additionally, we show that when $k=2$, these singular solutions are also solutions to the Born Infeld equation.