Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators

TL;DR

The paper proves that collapsed translating solitons in $\mathbb{R}^3 ext{ with finite entropy and finite genus converge, as } t o\pm\infty ext{, to a finite union of vertical planes, establishing a unique asymptotic configuration. It develops sharp PDE tools for drift-dominated equations on sausage-shaped domains, linking the drift Laplacian to the Yukawa equation and using potential-theoretic ideas to obtain removable singularities at infinity. A structure theorem shows infinity decomposes into standard regions (planes or grim reaper cylinders), yielding geometric classifications such as half-slab rigidity and entropy-two solitons. The work also constructs counterexamples to subsequential uniqueness in general noncompact settings and provides a robust framework for studying asymptotics along wings, culminating in precise classifications of translational solitons in half-slabs and entropy-two cases with empty forward limits.

Abstract

We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time $t\to\pm \infty$, any finite entropy, finite genus, embedded, collapsed translating soliton in $\mathbb{R}^3$ converges to a uniquely determined collection of planes. This requires global analysis of quasilinear soliton equations with non-perturbative drifts, which we analyze via sharp non-standard elliptic decay estimates for the drift Laplacian, implying improvements on the Evans-Spruck and Ecker-Huisken estimates in the soliton setting, and exploiting a link from potential theory of the Yukawa equation to heat flows with $L^\infty$-data on non-compact slice curves of these solitons. The structure theorem follows: such solitons decompose at infinity into standard regions asymptotic to planes or grim reaper cylinders. As one application, we classify collapsed translators of entropy two with empty limits as $t\to +\infty$.

Figures (10)

• Figure 1: Pitchfork mean curvature translating soliton in $\mathbb{R}^3$. (See Theorem B.)
• Figure 2: Two wings of a translator with finite entropy contained in a vertical slab. One of them is a planar wing (left) and the other one is a grim reaper-type wing (right.)
• Figure 3: The decomposition $\widetilde{\Sigma}$ for a pitchfork. You can appreciate three $\mathbf{e}_1$-graphs going up and one $\mathbf{e}_1$-graph going down. For translators with non-trivial genus, it is necessary to remove also a solid box containing the genus of $\Sigma.$
• Figure 4: Two different types of domains $U_j^{\rm up}$. The third type of domain, that we have not included in the picture, has a boundary whose non-polygonal part consists of the graph of a function $\varphi_j$ defined on the half-line $(-\infty,-t)$ and the horizontal half-line $\{x_3=0, x_2>t\}.$
• Figure 5: The domain $U_i^{\rm down}$.

Theorems & Definitions (115)

• Corollary
• Proposition 1.1: Doubling obstruction
• Definition 2.1
• Definition 2.2: Width of collapsed translators
• Theorem 2.3: entropy
• Lemma 2.4
• Proposition 2.5
• Proposition 2.6
• proof
• Proposition 2.7