Symmetry and Isoperimetry for Riemannian Surfaces

TL;DR

This paper introduces a scattering energy to quantify asymmetry in domains on convex surfaces, establishes sharp inequalities involving this energy, and characterizes symmetric domains, also providing a new proof of the sharp Sobolev inequality for Riemannian surfaces.

Contribution

It defines a novel scattering energy for Riemannian surface domains, links it to isoperimetric inequalities, and characterizes symmetric domains, along with a new proof of the Sobolev inequality.

Findings

Sharp quantitative isoperimetric inequalities involving scattering energy

Domains with zero scattering energy are convex and rotationally symmetric

New proof of the sharp Sobolev inequality 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 proof of the sharp Sobolev inequality for Riemannian surfaces which is independent of the isoperimetric inequality.