| Abstract | Let $k:\C\to \R$ be a smooth given function.
A $k$-loop is a closed curve $u$ in $\C$ having prescribed curvature $k(p)$ at every point $p\in u$.
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. |
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