The Limit of the Inverse Mean Curvature Flow on a Torus

Abstract

For an $H>0$ rotationally symmetric embedded torus $N_0\subset {\mathbb{R}}^3$, evolved by Inverse Mean Curvature Flow, we show that the total curvature $|A|$ remains bounded up to the singular time $T_{\max }$. We then show convergence of the $N_t$ to a $C^1$ rotationally symmetric embedded torus $N_{T_{\max }}$ as $t\to T_{\max }$ without rescaling. Later, we observe a scale-invariant $L^2$ energy estimate on any embedded solution of the flow in ${\mathbb{R}}^3$ that may be useful in ruling out curvature blowup near singularities in general.

15 pages, 2 figures, some minor typos corrected. Final version to appear in Proceedings of the American Mathematical Society

References in corpus (3)

• Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces
• Bounded Diameter Under Mean Curvature Flow
• Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces