Let $M^n \subset \mathbf{R}^{n+k}$ be a smoothly embedded minimal surface. When the dimension is $n = 2$ and the codimension is $k = 1$ the intersection of $M$ with planes is well understood. If $M$ and a plane $\Pi$ meet tangentially at a point $p \in M$, then locally near $p$, the intersection $M \cap \Pi$ is a union of finitely many smooth arcs meeting at $p$. (Unless $M = \Pi$ of course.) Moreover there is an even number of arcs, and they meet at equal angles; the number of arcs is determined by Almgren's frequency.
Is there a similar description for a tangential intersection $M \cap \Pi$, when the dimension of $M$ is arbitrary, but $\Pi$ remains two-dimensional?
