How close is too close for singular mean curvature flows?

TL;DR

The paper proves a sharp rigidity result for mean curvature flow near a multiplicity-one compact singularity: if two flows approach each other faster than any polynomial in the singular time, then they must coincide. The authors recast the problem in rescaled mean curvature flow (RMCF), express one flow as a small height graph over the other, and study the evolution of the height via a linear operator $L$ with a decaying error $E(u)$, using a frequency-energy framework to exclude nontrivial decays. A key contribution is a linear-with-errors result that yields a lower bound on the associated frequency and rules out super-exponential decay unless the height vanishes, enabling a backward-uniqueness conclusion. This extends prior results by showing that two distinct flows with the same compact tangent can be distinguished linearly unless one is the shrinking self-similar flow, thereby achieving a strong uniqueness statement for singular mean curvature flows. The techniques combine RMCF analysis, Gaussian-weighted energy methods, and Simon’s Lojasiewicz inequality in a setting where the flow is not itself self-similar.

Abstract

Suppose $(M^i_t)_{t\in [0,T)}$, $i=1,2$, are two mean curvature flows in $\mathbb{R}^{n+1}$ encountering a multiplicity one compact singularity at time $T$, in such a manner that for every $k$, the Hausdorff distance between the two flows, $d_H$, satisfies $d_{H}(M^1_t,M^2_t)/(T-t)^k \rightarrow 0$. We demonstrate that $M^1_t=M^2_t$ for every $t$. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where $M^1_t$ is itself a self-similarly shrinking flow.

Theorems & Definitions (11)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['L_with_errors']}
• proof : Proof of Theorem \ref{['main_theorem']}
• Lemma A.1