Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\mathscr{F} = \{ \phi: (F, \partial F) \to (M,K) : \phi \text{ is an embedding} \}$. By pulling back the metric we can talk about $\text{Area}(\phi)$ so we can ask if some element of $\mathscr{F}$ achieves the minimum area amongst all elements of $\mathscr{F}$. I suppose that I am mainly interested in knots in $S^3$ with the usual metric, if that simplifies things.

(1) Is this minimum achieved?

I have seen this paper where Proposition 1 might address my question - however, they allow for piecewise smooth maps. Is there any reason why the area minimizing surfaces would need to be smooth? This seems intuitively clear, but I am not familiar with how to prove such things.

(2) If the minimum is achieved, is it achieved by a genus minimizing surface? (I.E. the genus of $F$ is the genus of the knot).