Rigidity of ancient ovals in higher dimensional mean curvature flow

TL;DR

This work establishes a comprehensive rigidity and stability framework for compact ancient noncollapsed mean curvature flows in arbitrary dimensions, focusing on $k$-ovals with tangent flow $\mathbb{R}^{k}\times S^{n-k}(\sqrt{2(n-k)|t|})$. By proving a spectral uniqueness result in which $k$-ovals are uniquely determined (up to space-time rigid motion and parabolic dilation) by $(k-1)$ neutral-mode spectral ratios, and a spectral stability theorem that yields a Lipschitz, local parametrization of the moduli space, the authors demonstrate $\,\mathbb{Z}_2^{k}\times\mathrm{O}(n+1-k)$ symmetry and backward rigidity for these ancient solutions. The analysis introduces a novel test tensor to secure a quadratic almost concavity estimate in the collar region, handles nonradial terms arising from the inverse profile, and combines sharp asymptotics in three regions (parabolic, cylindrical, and soliton tip) with robust energy estimates in cylindrical and tip regions. The results extend rigidity phenomena known in low dimensions to general $k$ and $n$, supporting the conjecture that shrinking spheres and $k$-ovals exhaust compact ancient noncollapsed flows and providing key tools for the broader classification program of ancient geometric flows and related translators.

Abstract

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 $\mathbb{R}^{n+1}$, defined as ancient noncollapsed solutions whose tangent flow at $-\infty$ is given by $\mathbb{R}^k \times S^{n-k}((2(n-k)|t|)^{\frac{1}{2}})$ for some $k \in \{1,\dots,n-1\}$, 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 $\mathbb{R}^{n+1}$. We prove that $k$-ovals are $\mathbb{Z}^{k}_2 \times \mathrm{O}(n+1-k)$-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 $\mathbb{Z}^{k}_2 \times \mathrm{O}(n+1-k)$-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 $\mathbb{R}^4$, 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.

Theorems & Definitions (99)

• Definition 1.1: $k$-oval, c.f.DZ_spectral_quantization
• Definition 1.2: $\kappa$-quadraticity at $\tau_0$
• Theorem 1.3: spectral uniqueness
• Corollary 1.4: symmetry of $k$-ovals
• Theorem 1.5: spectral stability
• Theorem 1.6: quadratic almost concavity, CDDHS
• Theorem 1.7: uniform sharp asymptotics c.f. CDDHS, DZ_spectral_quantization
• Theorem 1.8: almost Gaussian collar
• Theorem 1.9: quadratic almost concavity
• Remark 1.10