Given a closed surface $S,$ does every essential subsurface contain a smooth curve? (curve = essential, simple, closed in the original surface) (essential subsurface = all boundary components are essential and curves)
The "worst case scenario", I'd imagine, is an annular subsurface where both boundary components are nowhere smooth. Without loss of generality, the annulus is $\epsilon-$thick.
I imagine the answer is yes, simply because there's enough "wiggle room". That's clearly not a proof, so I have some ideas/restatements below.
The above question is equivalent to: Can every continuous map $u:S^1\to \Sigma$ be approximated by a smooth map?
I'm familiar with some results in $\mathbb{R}^n,$ but I don't know how to apply them (or how to reduce the problem to a map that sends the circle to $\mathbb{R}^n$).
Many thanks!