Existence Results for the Nonlinear Hodge Minimal Surface Energy

Given a compact Riemannian manifold $(M^n,g)$ and a fixed cohomology class, $[\alpha^*] \in H^k(M)$, we consider the existence of a minimizer $\alpha \in [\alpha^*]$ of the generalized minimal surface energy $\int_M \sqrt{1+|\alpha|^2} dV_g$. When $k = 1$, we prove the existence of unique minimizers for every cohomology class $[\alpha^*]$. Next, when $k > 1$, we construct examples of singular solutions for finite cohomology class $[\alpha^*] \in H^k(S^k \times S^k,g)$, 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.