Remark 1.1.1 (The source editions).912105D5B712 References identify the original papers or the exact supplied editions. Citations distinguish printed page numbers from PDF page numbers and retain original theorem numbers. The source inventory records file hashes and which passages have been checked; an inventoried paper has not necessarily been read. Original PDFs and extracted text are not redistributed. Construction and classification results cited from the literature retain external proofs unless a calculation is supplied here.
§ 1.1 Sources and conventions35E9759B45E2
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MCF-stack collects special solutions of mean curvature flow in four categories: self-shrinkers, translators, self-expanders, and other special flows. This introduction and those four categories form the reading collection. Each example records its equation or construction, source-supported geometric properties, and a picture.
Remark 1.1.2 (Mean curvature flow and normalization).4EDA6AEDEA07 For a smooth immersion $X:M^ n\times I\to \mathbb R^ N$, mean curvature flow is the geometric equation
\[ (\partial _ tX)^\perp =\vec H, \]
where $\vec H$ is the trace of the second fundamental form and $\perp $ means normal projection. For hypersurfaces we use $H=\operatorname {div}_ M\nu $ and $\vec H=-H\nu $. Thus the outward normal of a round sphere has positive scalar mean curvature, and the sphere shrinks. For a smooth closed evolving submanifold,
\[ \frac{d}{dt}\operatorname {Area}(M_ t)=-\int _{M_ t}|\vec H|^2\, d\mu _ t. \]
Our self-shrinkers satisfy $\vec H=-X^\perp /2$ and generate $M_ t=\sqrt{-t}\, \Sigma $ for $t<0$. Self-expanders satisfy $\vec H=X^\perp /2$ and generate $M_ t=\sqrt t\, \Sigma $ for $t>0$. A translator satisfies $\vec H=V^\perp $ for a constant velocity $V$, with $M_ t=\Sigma +tV$. The translator pictures use upward unit speed unless stated otherwise. Dimensions, domains, scales, and time intervals are specified in each entry.
Remark 1.1.3 (Scope and illustrations).57833B199F67 The current collection contains 52 example families: 25 shrinkers, 13 translators, 6 expanders, and 8 other special flows. The last category includes stationary solutions, ancient solutions, and other distinguished evolutions. Ancient means defined for all $t<T$ for some finite $T$; eternal means defined for all real times. Neither condition implies that the solution is a soliton.
Explicit formulas are distinguished from implicit equations, ODE constructions, and asymptotic profiles. Every example has a figure. Interactive three-dimensional models allow rotation and zoom; higher-dimensional examples use projections, profiles, or schematics. Captions distinguish exact geometry, numerical ODE approximations, and schematic drawings, and identify truncated noncompact ends. These pictures do not replace the cited existence results. The collection is a growing reference, not an exhaustive classification or a formally verified library.
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