论文标题
古代渐近圆柱流的光谱量化
Spectral quantization for ancient asymptotically cylindrical flows
论文作者
论文摘要
我们研究古老的平均曲率流在$ \ mathbb {r}^{n+1} $中，其切线以$ - \ infty $为$ - \ infty $是一个收缩的圆柱体$ \ MATHBB {r}^{k} {k} {k} \ times s^{n-k} {n-k}（n-k}（n-k}（n-k}）我们证明，这些流的圆柱配置文件函数$ u $ u $具有渐近性$ u（y，ω，τ）=（y^\ top qy -2 \ 2 \ textrm {tr}（q））/|τ| + o(|τ|^{-1})$ as $τ\to -\infty$, where the cylindrical matrix $Q$ is a constant symmetric $k\times k$ matrix whose eigenvalues are quantized to be either 0 or $-\frac{\sqrt{2(n-k)}}{4}$.与Haslhofer获得的$ \ Mathbb {r}^{4} $中的气泡表量化定理相比，该定理在删除非汇总条件并有效的所有维度方面具有完全的一般性。此外，我们建立了对称性改进定理，该定理概括了Brendle-Choi的相应结果，第二作者是所有维度。最后，我们提供了两个定理的一些几何应用。特别是，我们获得了$ k $ ovals $ k $ ovals的$ \ mathbb {r}^{n+1} $的$ k $ ovals的$ \ textrm {o}（n-k+1）$对称性，它们是古老的非collapsed in CollyCollapsed Flow in $ \ Mathbb {r}^n+1} $满足的条件$ \ textrm {rk}（q）= k $，我们还获得了$ \ mathbb {r}^{n+1} $满足消失的排名条件的古代非收集流的分类。
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 $u(y,ω,τ)= (y^\top Qy -2\textrm{tr}(Q))/|τ| + o(|τ|^{-1})$ as $τ\to -\infty$, 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 $\textrm{rk}(Q)=0$.
