|Abstract||Let $k:Cto R$ be a smooth given function.
A $k$-loop is a closed curve $u$ in $C$ having prescribed curvature $k(p)$ at every point $pin u$.
We use variational methods to provide sufficient conditions for the existence of $k$-loops. Then we show that a breaking symmetry phenomenon may produce multiple $k$-loops, in particular when $k$ is radially symmetric and somewhere increasing.
If $k>0$ is radially symmetric and non increasing we prove that any embedded $k$-loop is a circle, that is, round circles are the only convex loops in $C$ whose curvature is a non increasing function of the Euclidean distance from a fixed point. The result is sharp, as there exist radially increasing curvatures $k>0$ which have embedded $k$-loops that are not circles.|