Symmetry and Isoperimetry for Riemannian Surfaces

For a domain $\Omega$ in a geodesically convex surface, we introduce a scattering energy $\mathcal{E}(\Omega)$, which measures the asymmetry of $\Omega$ by quantifying its incompatibility with an isometric circle action. We prove several sharp quantitative isoperimetric inequalities involving $\mathcal{E}(\Omega)$ and characterize the domains with vanishing scattering energy by their convexity and rotational symmetry. We also give a new proof of the sharp Sobolev inequality for Riemannian surfaces which is independent of the isoperimetric inequality.