A Gap Theorem for Willmore Tori and an application to the Willmore Flow

TL;DR

This paper establishes a gap theorem for Willmore tori, showing a quantifiable energy gap to the next critical point, and applies this to analyze the behavior of the Willmore flow, extending results to higher genus surfaces.

Contribution

The paper proves a new energy gap theorem for Willmore tori and applies it to the study of the Willmore flow, also extending the gap results to higher genus surfaces.

Findings

Existence of a quantifiable energy gap for Willmore tori.

Application of the gap theorem to the dynamics of the Willmore flow.

Extension of energy gap results to surfaces of higher genus.

Abstract

In 1965 Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $R^3$ is at least $2\pi^2$ and attains this minimal value if and only if the torus is a M\"obius transform of the Clifford torus. This was recently proved by Marques and Neves. In this paper, we show for tori there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.