The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

Abstract: We show that the intrinsic diameter of mean curvature flow in $\mathbb{R}<sup>3$ is uniformly bounded as one approaches the first singular time $T$. This confirms the \emph{bounded diameter conjecture} of Haslhofer. In addition, we establish several sharp quantitative estimates: the second fundamental form $A$ has uniformly bounded $L<sup>1$-norm on each time slice, $A$ belongs to the weak $L<sup>3$ space on the space-time region, and the singular set $\mathcal{S}$ has finite $\mathcal{H}<sup>1$-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. The proof is based on new covering theorems built from almost flat regions satisfying uniform content estimates. Furthermore, our arguments extend naturally to flows through singularities, yielding the same sharp estimates.