Existence Results for the Nonlinear Hodge Minimal Surface Energy

TL;DR

This paper investigates the existence of minimizers for a nonlinear minimal surface energy functional on Riemannian manifolds, proving existence and uniqueness for certain cases and constructing singular solutions in others.

Contribution

It establishes existence and uniqueness of minimizers for the energy functional when k=1 and constructs explicit singular solutions for higher cohomology classes, linking to the Born Infeld equation.

Findings

Unique minimizers exist for k=1 in each cohomology class.

Singular solutions are constructed for k>1 on specific manifolds.

For k=2, these solutions also solve the Born Infeld equation.

Abstract

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.