I'm trying to prove that if $\gamma$ is a Jordan curve (contained in the plane $\pi$) and $\Sigma\subset\mathbb{R}^3$ is a minimal surface with $\partial\Sigma=\gamma$, then $\Sigma$ must be the plane region $\Omega\subset\pi$ in $\pi$ delimited by $\gamma$.
It's clear that, if $\Sigma$ can be represented as a graph $u:\Omega\to\mathbb{R}$, then the only way to minimize the area is to chose $u\equiv 0$, so that $\Sigma=\Omega$.
But in the general case, I don't know how to justify it. I'm trying to use the maximum principle, but I don't know how.
Any tips?