MCF-stack: Special Solutions of Mean Curvature Flow

§ 4.11 Reported multiphase expanding surface constructionsBEE2570F7E0E

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. Liao — 2025-12-02 (first arXiv submission).
BEE2570F7E0E
55F8A309578D

Liao's 2025 preprint proposes self-expanding multiphase surfaces in $\mathbb R^3$ with regular triple curves and tetrahedral quadruple points, obtained from weighted-area minimization with prescribed asymptotic interface data. Triple sheets meet at $120$ degrees; four triple curves have the tetrahedral angle $\arccos (-1/3)$. The intended flow is $\sqrt t\, \Gamma $, $t>0$, sheetwise away from junctions. This entry records a reported construction, not a verified existence theorem of this catalogue. The compactness, persistence of junctions and conformal comparison used in Sections 3–5 require further checking. In particular one must distinguish the expander-weighted Euclidean metric from the hyperbolic metric; a conformal comparison does not automatically transport minimality or area minimization. Accordingly no definite global topology, embeddedness, completeness, uniqueness, or entropy assertion is assigned here. The tetrahedral junction drawing illustrates the local angle model only; that stationary cone by itself is not a new nontrivial self-expander. The panels are explicitly labeled proposed smoothing stages, rather than computed or proved flow slices.

Sources: [Liao2025].

Reported multiphase expanding surface constructions. Local tetrahedral junction model and proposed smoothing stages. This is a diagram of the preprint proposal, not a numerical or independently verified nontrivial expander.
Source-informed schematic. Local tetrahedral junction model and proposed smoothing stages. This is a diagram of the preprint proposal, not a numerical or independently verified nontrivial expander. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Reported multiphase construction in ℝ³ · Euclidean space ℝ³ · Not recorded
Ambient space
Euclidean space ℝ³
Topology
Not certified from the current review
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Not specified for this subfamily
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
Not recorded for the full parameter manifold.
Entropy
Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area
Not recorded: No quantitative conclusion is recorded from the cited construction.

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References on this page

  1. [Liao2025] Wei-Hung Liao. Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions. arXiv preprint, 2025. URL: https://arxiv.org/abs/2512.02578, arXiv:2512.02578.All uses in this project ↩ 1
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