The Willmore Flow of Tori of Revolution

Abstract: This is a joint work with Anna Dall'Acqua, Adrian Spener and Reiner Schätzle.

We study the $\textcolor{red}{\textbf{Willmore flow}}$ of tori that have a revolution symmetry - so-called tori of revolution. Luckily, the Willmore flow preserves this symmetry. Because of that we can look at the flow as an evolution of the "profile curves" - a reduction of the dimension!

We will examine the geometry of this curve evolution and understand why it is somewhat natural to look at those curves in $\textcolor{red}{\textbf{hyperbolic geometry}}$. We prove:

$\textcolor{green}{ \textbf{If the hyperbolic length of the profile curves remains bounded, then the Willmore flow converges.}}$

The remaining question: How can the hyperbolic length of those curves be controlled? We use variational methods to $\textcolor{red}{\textbf{control the hyperbolic length}}$ by the Willmore energy - but this control is only available below an energy level of $\textcolor{red}{\mathbf{8\pi}}$. We obtain:

$\textcolor{green}{\textbf{If we start the Willmore flow with a torus of revolution of Willmore energy below $8\pi$, then the flow converges}.}$

If time allows: The threshold of $8\pi$ is also sharp and plays an important role in the context of the Willmore functional. It is also the same threshold that was already found by E. Kuwert and R. Schätzle for the Willmore flow of spheres.

analysis of PDEsdifferential geometry

Audience: researchers in the discipline

( paper )

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