2022/11/04 by Wenkui Du, Du, Wenkui, Jingze Zhu +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2211.02595
openalex publication_date 2022/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study ancient mean curvature flows in ℝn+1 whose tangent flow at -∞ is a shrinking cylinder ℝk× Sn-k(√(2(n-k)|t|)), where 1≤ k≤ n-1. We prove that the cylindrical profile function u of these flows have the asymptotics u(y,ω,τ)= (y^\top Qy -2\textrmtr(Q))/|τ| + o(|τ|-1) as τ→ -∞, where the cylindrical matrix Q is a constant symmetric k× k matrix whose eigenvalues are quantized to be either 0 or -(√(2(n-k)))/(4). Compared with the bubble-sheet quantization theorem in ℝ4 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 \textrmO(n-k+1) symmetry of k-ovals in ℝn+1 which are ancient noncollapsed flows in ℝn+1 satisfying full rank condition that \textrmrk(Q)=k, and we also obtain the classification of ancient noncollapsed flows in ℝn+1 satisfying vanishing rank condition that \textrmrk(Q)=0.