Suppose $\Gamma\subset\mathbb{R}^3$ is a smooth, embedded, closed curve and suppose $n(p)\in N_p\Gamma$ is a smooth vector field on $\Gamma$ and normal to $\Gamma$. Does there always exist a surface $\Sigma\subset\mathbb{R}^3$ such that $\partial\Sigma=\Gamma$ and the inner co-normal* to the surface, $\nu(x)=n(x)$ for $x\in\partial\Sigma$?
Without the prescribed conormal condition, the answer is certainly yes. You could just solve the plateau problem (which also gives you an area-minimizing surface in addition). I suppose you could also get a cone by connecting every point of the $\Gamma$ to some fixed point in $\mathbb{R}^3$ and then smoothen to get the existence of such a surface.
But when you add the prescribed conormal, is the problem over-determined? Or is it still possible to get such a surface?
*The inward-pointing vector field that is normal to the boundary and tangential to the surface.
One way of constructing such an immersed surface would be to first 'flow' the boundary curve, $\Gamma$ along the field $n(x)$ for as long as injectivity is preserved i.e., define the map $\phi:(s,t)\mapsto t n(\Gamma(s))$ and show that there is some $\tau>0$ such that the map is injective for $t\leq\delta$. This defines a 'ribbon-like' surface, $\tilde\Sigma$, whose boundary lies on $\Gamma$.
Now, let $\tilde\Gamma$ be the curve given by $f(s,\delta)$ (the other boundary of the ribbon). Choose some point $x_0\in\mathbb{R}^3$ away from the $\tilde\Sigma$ and consider the cone $C$ that connects $\tilde\Gamma$ to $x_0$. The desired surface can be constructed by gluing $C$ to $\tilde\Sigma$ and mollifying with a function whose support contains $\tilde\Gamma$, but not $\Gamma$.