2025/04/13 by Beomjun Choi, Choi, Beomjun, Wenkui Du +3 · 2 citations
Computer Science · Engineering · Mathematics · #35A21 #35K55 #53E10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #History and Theory of Mathematics #Image Processing and 3D Reconstruction #Mechanics and Biomechanics Studies
paper · pdf · doi:10.48550/arxiv.2504.09741
openalex publication_date 2025/04/13 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
In this paper, we consider the classification of compact ancient noncollapsed mean curvature flows of hypersurfaces in arbitrary dimensions. More precisely, we study k-ovals in ℝn+1, defined as ancient noncollapsed solutions whose tangent flow at -∞ is given by ℝk × Sn-k((2(n-k)|t|)(1)/(2)) for some k ∈ \1,…,n-1\, and whose fine cylindrical matrix has full rank. A significant advance achieved recently by Choi and Haslhofer suggests that the shrinking n-sphere and k-ovals together account for all compact ancient noncollapsed solutions in ℝn+1. We prove that k-ovals are ℤk2 × O(n+1-k)-symmetric and are uniquely determined by (k-1)-dimensional spectral ratio parameters. This result is sharp in view of the (k-1)-parameter family of ℤk2 × O(n+1-k)-symmetric ancient ovals constructed by Du and Haslhofer, as well as the conjecture of Angenent, Daskalopoulos and Sesum concerning the moduli space of ancient solutions. We also establish a new spectral stability theorem, which suggests the local (k-1)-rectifiability of the moduli space of k-ovals modulo space-time rigid motion and parabolic rescaling. In contrast to the case of 2-ovals in ℝ4, resolved by Choi, Daskalopoulos, Du, Haslhofer and Sesum, the general case for arbitrary k and n presents new challenges beyond increased algebraic complexity. In particular, the quadratic concavity estimates in the collar region and the absence of a global parametrization with regularity information pose major obstacles. To address these difficulties, we introduce a novel test tensor that produces essential gradient terms for the tensor maximum principle, and we derive a local Lipschitz continuity result by parameterizing k-ovals with nearly matching spectral ratio parameters.