Rigidity of stationary and uniformly rotating planar Euler flows in the $C^1$ Yudovich class

We prove a rigidity theorem for stationary and uniformly rotating solutions of the two-dimensional incompressible Euler equation with vorticity $ ω_0\in C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2). $ If the angular velocity $Ω$ satisfies \[ Ω\leq\frac12\inf_{\math