Let $(M_j \mid j \in \mathbf{N})$ be a sequence of compact minimal surfaces in the unit ball $B \subset \mathbf{R}^3$, with boundaries $\partial M_j \subset \partial B$. Assume that the sequence converges to the horizontal plane $\mathbf{R}^2 \times \{ 0 \}$ with multiplicity $Q \geq 2$: \begin{equation} \lvert M_j \rvert \to Q \cdot \lvert \mathbf{R}^2 \times \{ 0 \} \rvert \quad \text{ as $j \to \infty$} \end{equation} in the varifold topology. Note that no bound for the Morse index or genus of the surfaces is assumed: $\mathrm{genus} \, M_j \to \infty$ and $\mathrm{index} \, M_j \to \infty$ is not forbidden in principle.
Question. Does the theory developed by Colding and Minicozzi give any further information about the convergence? If not, what kind of sheeting theorem is available when a uniform bound is assumed for the genus?
My (extremely limited) understanding of their work is that it gives decomposition results. Loosely I've heard the results described as stating that minimal surfaces can be decomposed into portions that resemble catenoidal necks, multi-valued graphs modelled on helicoids and single-valued graphs. (I don't actually know how accurate this is, even on a heuristic level.)
I would imagine that when there is a uniform bound like \begin{equation} \mathrm{genus} \, M_j \leq C \quad \text{ for all $j$}, \end{equation} then this could be applied to give a kind of sheeting theorem—perhaps multivalued—, valid away from the points where the necks concentrate. If the genus goes to infinity then this could not hold, but perhaps some other conclusions could be drawn on the sequence.
