Given a (smooth) simple closed curve $C \subset \mathbb{R}^3$, is there a (smooth) surface $S$ with $\partial S = C$? I'm aware there is a variational problem to find among such surfaces the one with minimal area. Here I'm interested in the statement and proof of some existence theorem. Some cases where the existence is clear: if $C$ is planar, and more generally if $C$ is the curve of intersection of a a graph and a cylinder, $x_1=g(x_2,x_3)$ and $f(x_2,x_3)=0$
(This question came up when I read in a calculus book that $\text{curl }\mathbf{F}=\mathbf{0}$ implies $\mathbf{F}=\nabla f$ for some scalar function $f(x,y,z)$. The proof was: $\mathbf{F}=\nabla f$ iff $\mathbf{F}$ is conservative; to show $\mathbf{F}$ is conservative, consider $\int_C \mathbf{F}\cdot d\mathbf{r}$ for any closed curve $C$. "THERE IS" a surface $S$ with $C$ as its boundary. By stokes theorem, and the fact that $\text{curl }\mathbf{F}=0$, $\int_C \mathbf{F}\cdot d\mathbf{r} = 0$. So, my question is why this surface even exists.)
As already stated by Cheerful Parsnip, such surfaces exist.
In fact, you can choose them to be compact and oriented (then they are Seifert surfaces). You can find an elementar proof in [Saveliev - Lectures on the Topology of 3-Manifolds].
If you know a little bit more machinery from differential topology, this (much sexier) argument will do the trick.