Planar loops with prescribed curvature: Existence, multiplicity and uniqueness results

Let k: ℂ → 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 Ie ∈ 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.