Spectral quantization for ancient asymptotically cylindrical flows

TL;DR

This paper investigates ancient mean curvature flows with cylindrical tangent flows, proving spectral quantization of their profile functions, establishing symmetry improvements, and applying these results to classify and analyze ancient flows and geometric objects.

Contribution

It extends spectral quantization results to all dimensions without noncollapsing assumptions and generalizes symmetry improvement theorems.

Findings

Spectral quantization of cylindrical matrix Q with eigenvalues 0 or -√(2(n-k))/4.

Asymptotic behavior of profile functions in ancient flows.

Classification and symmetry results for ancient noncollapsed flows.

Abstract

We study ancient mean curvature flows in $\mathbb{R}^{n+1}$ whose tangent flow at $-\infty$ is a shrinking cylinder $\mathbb{R}^{k}\times S^{n-k}(\sqrt{2(n-k)|t|})$, where $1\leq k\leq n-1$. We prove that the cylindrical profile function $u$ of these flows have the asymptotics $u(y,\omega,\tau)= (y^\top Qy -2\textrm{tr}(Q))/|\tau| + o(|\tau|^{-1})$ as $\tau\to -\infty$, where the cylindrical matrix $Q$ is a constant symmetric $k\times k$ matrix whose eigenvalues are quantized to be either 0 or $-\frac{\sqrt{2(n-k)}}{4}$. Compared with the bubble-sheet quantization theorem in $\mathbb{R}^{4}$ obtained by Haslhofer and the first author, this theorem has full generality in the sense of removing noncollapsing condition and being valid for all dimensions. In addition, we establish symmetry improvement theorem which generalizes the corresponding results of Brendle-Choi and the second author to all dimensions. Finally, we give some geometric applications of the two theorems. In particular, we obtain the asymptotics, compactness and $\textrm{O}(n-k+1)$ symmetry of $k$-ovals in $\mathbb{R}^{n+1}$ which are ancient noncollapsed flows in $\mathbb{R}^{n+1}$ satisfying full rank condition that $\textrm{rk}(Q)=k$, and we also obtain the classification of ancient noncollapsed flows in $\mathbb{R}^{n+1}$ satisfying vanishing rank condition that $\textrm{rk}(Q)=0$.