Control of the isoperimetric deficit by the Willmore deficit

TL;DR

This paper demonstrates that for smoothly embedded sphere-like surfaces, the deviation from optimal isoperimetric ratio can be effectively bounded by the Willmore energy difference.

Contribution

It establishes a new quantitative relationship linking the isoperimetric deficit to the Willmore deficit for embedded surfaces of sphere type.

Findings

Isoperimetric deficit is controlled by Willmore deficit.

Provides bounds relating geometric deficits.

Focuses on smooth embedded sphere-like surfaces.

Abstract

In the class of smoothly embedded surfaces of sphere type we prove that the isoperimetric deficit can be controlled by the Willmore deficit.