Colding and Minicozzi proved the following limit lamination theorem (see Theorem 8.26 in "A course in Minimal Surfaces" by Colding and Minicozzi for the complete statement)
THEOREM: Let $\Sigma_i \subset B_{R_i}(0) \subset \mathbb{R}^3$ be a sequence of embedded minimal disks with $\partial \Sigma_i \subseteq \partial B_{R_i}$ where $R_i \rightarrow \infty$. If $\sup_{B_1 \cap \Sigma_i}|A|^2 \rightarrow \infty$, then there exists a subsequence, $\Sigma_j$ and a Lipschitz curve $\mathcal{S} \colon \mathbb{R} \to \mathbb{R}^3$ s.t. after rotation of $\mathbb{R}^3$ for each $1 > \alpha > 0$, $\Sigma_j \setminus \mathcal{S}$ converges in the $C^\alpha$-topology to the foliation $\mathcal{F} = \{x_3 = t\} $ of $\mathbb{R}^3$.
Do I get the same conclusion if I assume the surfaces $\Sigma_i$ to be minimal with respect to a Riemannian metric $g_i$ on $B_{R_i}$ such that $g_i$ converges to the flat Euclidean metric in the $C^\infty_{loc}$ topology?
