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

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 Möbius 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.

9 pages. In this new version we performed some small changes to improve the exposition. To appear in Nonlinear Analysis: Theory Methods & Applications

Cited by in corpus (4)

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• The Willmore Flow of Tori of Revolution
• Computational p-Willmore Flow with Conformal Penalty
• Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization