Symmetry and isoperimetry for Riemannian surfaces

Abstract

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 of the sharp Sobolev inequality for Riemannian surfaces.