MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.15 Annuloid translators in slabs (Gama–Martín–Møller, 2022; Hoffman–Martín–White, 2023)93A7D23F7FA3

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. Gama–Martín–Møller — 2022-09-04 (first arXiv submission). Background only: General entropy relation used for annuloids in the later construction paper; not a reference constructing this family.
  2. Hoffman–Martín–White — 2023-08-04 (first arXiv submission).
93A7D23F7FA3
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For every inner half-width $b\geq \pi /2$ and necksize $s>0$, Hoffman–Martín–White construct a complete properly embedded annular translator in $\mathbb R^3$. Their necksize is the distance from the $z$-axis to $M\cap \{ y=0\} $; it is not assumed to be a circular neck radius. The surface lies in a slab $|y|\leq B$, has both vertical coordinate-plane reflection symmetries, and is nonrotational. Its topology is $S^1\times \mathbb R$: genus zero and two ends.

After reflecting the source's downward translator, $M_ t=M+te_3$ for all $t\in \mathbb R$. The positive-height asymptotics are four planes $y=\pm b,\pm B$, counted with multiplicity if $b=B$; the negative-height end of the ambient space sees no surface. Four asymptotic planes in one height direction do not mean four topological ends. The outer half-width $B$ is an output of the construction, not an independent prescribed parameter. Capped and uncapped behaviors occur. No closed formula or uniqueness for fixed $(b,s)$ is supplied.

The entropy is exactly $4$, by [HMWAnnuli24], Remark 7.9, using [GMM25], Corollary 8.5. This is not a fixed-center Gaussian-area identity. Existence and the parameters are in Theorem 1.1, Definition 1.2, and Sections 7 and 22 of [HMWAnnuli24]. The displayed neck model is heuristic.

Annuloid translators in slabs. Heuristic capped annuloid: two strip-like sheets joined around a neck. The shape is not computed from the translator PDE, and no pictured neck size specifies the paper’s parameters.
Heuristic schematic. Heuristic capped annuloid: two strip-like sheets joined around a neck. The shape is not computed from the translator PDE, and no pictured neck size specifies the paper’s parameters. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
Annuloid · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Annulus S¹ × ℝ
Embeddedness
Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness
Noncompact
Surface genus
0
Topological ends
2, despite four vertical asymptotic planes
Entropy
Exact formula: λ = 4 (Hoffman–Martín–White, Remark 7.9).
Gaussian area
Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

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References on this page

  1. [HMWAnnuli24] David Hoffman, Francisco Martín, and Brian White. Translating annuli for mean curvature flow. 2024. arXiv:2308.02210v4. Selected theorem statements and construction passages checked November 6, 2026; proofs are cited, not independently verified. URL: https://arxiv.org/abs/2308.02210v4.All uses in this project ↩ 1 ↩ 2
  2. [GMM25] Eddygledson Souza Gama, Francisco Martín, and Niels Martin Møller. Finite entropy translating solitons in slabs. 2025. arXiv:2209.01640v3. Selected theorem statements and construction passages checked November 6, 2026; proofs are cited, not independently verified. URL: https://arxiv.org/abs/2209.01640v3.All uses in this project ↩ 1
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