I've been thinking about topologies on the spaces of k-rectifiable sets (Hausdorff metric topology, varifold topology, etc) in $\mathbb{R}^n$, and I'm wondering if there are hypotheses under which they imply with $C^k$ convergence away from singularities.
For example, if $\{v_i\}$ is a sequence of k-rectifiable sets with uniformly bounded mass and curvature, and $v_i \rightarrow v$ in the Hausdorff metric, the Allard Compactness Theorem implies that it converges to the same limit in the varifold sense. Are there any additional hypotheses on a Hausdorff convergent sequence that would imply $C^k$ (or perhaps smooth) convergence away from singularities?
Consider the one-dimensional case. Let $\{v_i\}$ be a sequence of $C^{k+1}$ arcs (closed, connected 1-manifolds with boundary) in $\mathbb{R}^n$ converging to a $C^k$ arc in the Hausdorff sense, and suppose the first $k+1$ derivatives of the unit speed parametrizations of the $\{v_i\}$ are uniformly bounded (there should be a more natural way to say this). The mean value inequality implies equicontinuity of the first k derivatives, so we get a $C^k$ convergent sub-sequence to a $C^k$ function by Arzela-Ascoli. This must be the same as the Hausdorff limit arc. I think this can be extended to 1-varifolds composed of finitely many arcs, with appropriate conditions at the singularities.
If this is correct, does it generalize to higher dimensions? (Related question: is $C^\infty(M,R^n)$ a nuclear space, where $M$ is a compact manifold with boundary?)