Symmetry and Isoperimetry for Riemannian Surfaces

Joseph Ansel Hoisington, Peter McGrath

Published 2020-06-15Version 1

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.