On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is H² H2 -close to a multiply covered circle and is sufficiently rotationally symmetric.