The regularity of intersection of a minimal surface and a surface of positive mean curvature?

Question

Let $(M^3,g)$ be a closed smooth Riemannian manifold. Let $\Sigma_1$ be a closed minimal surface and $\Sigma_2$ a closed surface of positive mean curvature in $M^3$. Set $A=\Sigma_1\cap\Sigma_2$. If $A\neq\emptyset$, then what is the regulariy of $A$? Obviously, $A$ is closed and has no interior point. Can we say something more? For example, what is largest Hausdorff dimension can $A$ have? Can $A$ be a union of several arcs and isolated points?