The Limit of the Inverse Mean Curvature Flow on a Torus

TL;DR

This paper investigates the inverse mean curvature flow of rotationally symmetric tori in three-dimensional space, demonstrating bounded curvature up to singularity and convergence to a limit torus, with implications for understanding singularity formation.

Contribution

It establishes bounded total curvature for the flow on tori and proves convergence to a smooth limit, introducing a scale-invariant energy estimate relevant for singularity analysis.

Findings

Total curvature remains bounded up to singularity.

Flow converges to a $C^{1}$ rotationally symmetric torus.

Introduces a scale-invariant $L^{2}$ energy estimate.

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 $t \rightarrow T_{\max}$ 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.