The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow
Yiqi Huang, Wenshuai Jiang
TL;DR
This work resolves Haslhofer’s bounded diameter conjecture for mean curvature flow in 3D by proving a uniform bound on the intrinsic diameter up to the first singular time, without convexity assumptions. It introduces a comprehensive framework of almost regular flows and uses entropy-guided, quantitative splitting together with cylindrical coverings to obtain optimal curvature and regularity estimates, including a uniform $L^1$ bound for the second fundamental form and a sharp weak $L^3$ bound in space-time. The authors show that the final space-time singular set has finite 1D parabolic measure and is 1-rectifiable, with a refined Minkowski-type control near the singular set. Their covering-theorem approach avoids reliance on canonical neighborhood theorems, extends through singularities, and yields sharp structural results that also illuminate potential extensions to other geometric flows such as the Ricci flow.
Abstract
We show that the intrinsic diameter of mean curvature flow in $\mathbb{R}^3$ is uniformly bounded as one approaches the first singular time $T$. This confirms the bounded diameter conjecture of Haslhofer. In addition, we establish several sharp quantitative estimates: the second fundamental form $A$ has uniformly bounded $L^1$-norm on each time slice, $A$ belongs to the weak $L^3$ space on the space-time region, and the singular set $\mathcal{S}$ has finite $\mathcal{H}^1$-Hausdorff measure. All of the results are optimal due to the marriage ring example and our results do not require any convexity assumptions on the surfaces. Furthermore, our arguments extend naturally to flows through singularities, yielding the same sharp estimates.