Calculus of Variations and Geometric Measure Theory

A. Mondino - H. T. Nguyen

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

created by mondino on 20 Aug 2013
modified on 20 Feb 2014


Nonlinear Analysis: Theory Methods & Applications

Inserted: 20 aug 2013
Last Updated: 20 feb 2014

Year: 2013


In 1965 Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in ${\mathbb R}^3$ 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.