In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $Omega$, such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.
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