UNIQUENESS OF NONCOLLAPSED TRANSLATORS IN MEAN CURVATURE FLOW = 평균 곡률 흐름에서 비붕괴 평행이동 해의 유일성

다국어 초록 (Multilingual Abstract)

In this paper, we consider the classification of noncollapsed translating mean curvature flows in $\mathbb{R}^{n+1}$ whose level sets are compact. More precisely, we study full-rank translators in $\mathbb{R}^{n+1}$ satisfying $|t|^{-1/2}M_t\rightarrow \mathbb{R}^k\times\mathbb{S}^{n-k}(\sqrt{2(n-k)})\ \text{in}\ C^{\infty}_{loc}$ as $t\rightarrow-\infty$, for some $k \in \{1,\dots,n-1\}$, and whose fine cylindrical matrix has one-dimensional kernel, which, as a consequence of being a translator, is automatically spanned by the translating direction. A significant advance achieved recently by Choi and Haslhofer (CH24-classification of ancient noncollapsed flows in $\mathbb{R}^4$)suggests that under strictly convex assumption — which is a mild assumption, since strict convexity always arises after finitely many line-splittings — the dimension of kernel of the fine cylindrical matrix must be either 0 or 1, and in the latter case the solution is necessarily a translator. Consequently, every noncollapsed, strictly convex, noncompact ancient mean curvature flow must be a full-rank translator.\par \medskip
The central result of this paper is that full-rank translators are $\mathbb{Z}^{k-1}_2 \times \mathrm{O}(0\times\mathbb{R}^{n+1-k})$-symmetric and are uniquely determined by $(k-2)$-dimensional spectral ratio parameters, with both the symmetry and the parameters naturally arising from the geometry of their level sets. This result can be viewed as the application of the CDZ25 (rigidity of ancient ovals in higher dimensional mean curvature flow) to the translator case, as well as a the higher dimensional generalization of CHH23 (classification of noncollapsed translators in $\mathbb{R}^4$). In contrast to the case of full-rank translator in $\mathbb{R}^4$, resolved in CHH23, 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. Nevertheless, many of these challenges can be addressed by exploiting natural analogies with the CDZ25 framework.\par \medskip
We begin by establishing uniform sharp asymptotics together with $O(0\times\mathbb{R}^{n-k+1})$-symmetry, which provides the natural setup for the three-region (cylindrical, soliton, and collar) analysis. We then prove the almost Gaussian collar by using an almost-quadratic concavity estimate for a test tensor that supplies the gradient terms needed for the tensor maximum principle. This tensor is obtained by adding the $\frac{1}{Q}$ correction term from CHH23—reflecting the discrepancy between the level-set geometry and the global flow—to the CDZ25 tensor.\par \medskip
With these ingredients in place, we establish the spectral uniqueness via the three-region energy estimates, which in turn yields the central classification result. The analysis in the cylindrical and soliton regions is relatively straightforward, as their geometry is nicely determined by the sharp asymptotics. In contrast, the collar region presents genuine difficulty due to its a priori undetermined geometry. Here we use the almost Gaussian collar to show that the difference between two flows in the collar region can be controlled by the translation region—that is, the portion of the collar overlapping with the cylindrical region.

목차 (Table of Contents)

• 1. Sharp asymptotics and circular symmetry 2. Quadratic Almost Concavity 3. Spectral Uniqueness Theorem