MCF-stack: Special Solutions of Mean Curvature Flow

MCF-stack: Special Solutions of Mean Curvature Flow

A growing Stacks-style reference collecting self-shrinkers, translators, self-expanders, ancient solutions, and other special solutions of mean curvature flow.

312 entries · 5 chapters · 155 sections · 1 manuscript · 137 references

Each catalogue entry records formulas or construction references, source-supported properties, and a picture. Drawings distinguish exact geometry, numerical ODE approximations, projections, and schematics.

Browse by properties →Ambient space · Topology, genus and ends · Embeddedness · Gaussian area and entropy

What is mean curvature flow?

Mean curvature flow moves a curve, surface, or higher-dimensional submanifold in the direction of its mean curvature vector. For an immersion \(X:M^n\times I\to\mathbb R^{n+1}\), the geometric evolution equation is

\[(\partial_t X)^\perp=\mathbf H.\]

Here \(\mathbf H\) is the mean curvature vector and \(\perp\) denotes projection onto the normal space. With an outward unit normal \(\nu\) and scalar mean curvature \(H=\operatorname{div}_M\nu\), we use \(\mathbf H=-H\nu\). A round sphere therefore shrinks under the flow.

The flow is the gradient flow of area: it decreases area for smooth closed submanifolds, but it can develop singularities. Special solutions provide explicit examples and models for understanding the geometry near singularities and at large scales.

What this stack collects

  • Self-shrinkers: solutions that contract by dilation, normalized by \(\mathbf H+X^\perp/2=0\).
  • Translators: solutions that move with a constant velocity \(V\), satisfying \(\mathbf H=V^\perp\).
  • Self-expanders: solutions that expand by dilation, normalized by \(\mathbf H-X^\perp/2=0\).
  • Other special solutions: ancient and eternal flows, stationary minimal submanifolds, and rotating or more general self-similar solutions.

An ancient solution exists for all times before some finite time; an eternal solution exists for all real times. These time domains do not by themselves imply self-similarity.

Following the idea of the Stacks Project, MCF-stack aims to connect examples, constructions, properties, and classification results through stable tags, cross-references, and precise source attribution. This is a growing reference, with hypotheses and proof status stated explicitly.

Selected examples

The reading collection consists of an introduction and four categories: shrinkers, translators, expanders, and other special flows. Each example includes a figure and precise references.

Table of contents

Chapter 1. Introduction358B9256A93C
  1. 1.1 Sources and conventions35E9759B45E2
Chapter 2. Shrinkers766061837311

ℝ²

  1. 2.1 Closed planar shrinking curves (Abresch–Langer, 1986; Halldórsson, 2010; Drugan–Lee–X. H. Nguyen, 2017; Colding–Minicozzi II, 2019)B4D1BD157D4B
  2. 2.2 Spoon-shaped shrinking network (Chen–Guo, 2007; Chang, 2021)E8ED05753059
  3. 2.3 Shrinking lens network (Chen–Guo, 2007; Chang, 2021)5B73D42E7202
  4. 2.4 Shrinking fish network (Chen–Guo, 2007; Chang, 2021)BAA362BD1D2D
  5. 2.5 Three-ray shrinking star (Chen–Guo, 2007; Chang, 2021)66EFFE766DF2
  6. 2.6 Four-ray shrinking star (Chen–Guo, 2007; Chang, 2021)D3FA350D95E8
  7. 2.7 Five-ray shrinking star (Chen–Guo, 2007; Chang, 2021)429D29F97379
  8. 2.8 Shrinking rocket network (Chen–Guo, 2007; Chang, 2021)C8F779619BD0
  9. 2.9 Round shrinking spheres (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)2643D15A6912
  10. 2.10 Planes through the shrinker center (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)CB99B1567911
  11. 2.11 Nonclosed shrinking curves dense in an annulus3A8C87C9FFF8
  12. 2.12 Cisgeminate-eye shrinkerAAF58D62FB97

ℝ³

  1. 2.13 Embedded rotational shrinking tori (Angenent, 1992; Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017; Berchenko-Kogan, 2018)EB7B6435F145
  2. 2.14 Chopp cubical shrinker candidate90FE05D9053E
  3. 2.15 Chopp octahedral shrinker candidateEAAEF7E5907F
  4. 2.16 Chopp punctured-torus saddle candidate753FE61621D7
  5. 2.17 Chopp two-handled cone candidate50368E5C98D8
  6. 2.18 Numerical four-handled cone shrinker candidate (Chopp, 1994; Angenent–Ilmanen–Chopp, 1995)5684D97D637F
  7. 2.19 Generalized shrinking cylinders1A070FD35EB2
  8. 2.20 Rotational shrinking trumpet ends72885E0FF22A
  9. 2.21 Nguyen's one-ended shrinking saddlesEDE40E8E1379
  10. 2.22 One-ended high-genus shrinkers from sphere–plane gluing (Kapouleas–Kleene–Møller, 2011; Sun–Zhichao Wang, 2018)2285B0C987A6
  11. 2.23 Closed shrinkers of large even genus via torus–sphere gluing (Colding–Minicozzi II, 2009; Møller, 2011; Drugan–Lee–X. H. Nguyen, 2017)1AD8ABE5C210
  12. 2.24 Immersed rotational shrinking spheres (Drugan, 2013; Drugan–Lee–X. H. Nguyen, 2017)2101D51F1E32
  13. 2.25 Drugan–Kleene immersed shrinking familiesF2D20223E5E5
  14. 2.26 Shrinking ends asymptotic to isoparametric cones33FBBB16A0B1
  15. 2.27 Ketover shrinkers with Platonic symmetryB675C6112328
  16. 2.28 Variational embedded shrinking tori (Drugan–X. H. Nguyen, 2017; Chu–Sun, 2023)7E48D8C7D552
  17. 2.29 Lens-shaped shrinking surface clusters82C42A1C42B4
  18. 2.30 Closed shrinkers from spherical doublingEB175520B015
  19. 2.31 Sun–Wang–Zhou shrinkers of arbitrarily large entropy879F42A66D65
  20. 2.32 Noncompact shrinkers of every positive genus (Buzano–H. T. Nguyen–Schulz, 2021; Chu–Sun, 2023)0E02104583BA
  21. 2.33 Chu–Sun genus-one shrinkers below the shrinking doughnutB050FA22AB2D
  22. 2.34 Ilmanen–White two-ended shrinkersC7322187A6FC
  23. 2.35 Ketover self-shrinking doubled planes8423ACEAF2E2
  24. 2.36 Flow-generated Platonic and polygonal shrinkers1AF8BDC11238
  25. 2.37 Flow-generated sphere–plane shrinkers036F44595178
  26. 2.38 Flow-generated three-ended shrinkers5245200FA4A7
  27. 2.39 Shao–Zou shrinkers with any prescribed number of endsEB2734BB2E46

2 families from lower dimensions →

ℝ⁴

  1. 2.40 Products of closed shrinking curves44DD8966BC9A
  2. 2.41 Higher-codimension spherical shrinkers (Smoczyk, 2005; Sun–Zhu, 2020)6C99F82AA214
  3. 2.42 An embedded self-shrinking Möbius bundle (Lee–Mu-Tao Wang, 2007; Braxton–Lee–Zhu, 2026)5D9025FCD9CD
  4. 2.43 Compact periodic Lagrangian quadric shrinkers6AFBAD1A608F
  5. 2.44 Noncompact periodic Lagrangian shrinkers9B86DFDA9217
  6. 2.45 The Clifford shrinking torus in four-spaceCDFAEA3DDEF8
  7. 2.46 Cylinders over embedded rotational shrinking tori (Angenent, 1992; Berchenko-Kogan, 2018; Chen–Lee–Sun–Zhao, 2026)99055D91C188
  8. 2.47 Park bi-rotational shrinking ends51C4E786B6C8
  9. 2.48 Complete bi-rotational shrinking continuationsFDFAD175AF52

10 families from lower dimensions →

Higher dimensions (N ≥ 5)

  1. 2.49 Lagrangian shrinkers with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)689C0F332420
  2. 2.50 Lee–Wang embedded quadric shrinkersA400A6F0174B
  3. 2.51 Riedler shrinkers from isoparametric foliationsEDE4E49195FA

17 families from lower dimensions →

Curved ambient spaces

  1. 2.52 Yamamoto Lagrangian shrinkers in Calabi–Yau cones26AC43F0ACEF
Chapter 3. Translators2760875656EF

ℝ²

  1. 3.1 The grim reaper and its cylinders (Hoffman–Ilmanen–Martín–White, 2018; Hoffman–Ilmanen–Martín–White, 2019)505486401C44

ℝ³

  1. 3.2 The rotational bowl translator (Clutterbuck–Schnürer–Schulze, 2005; Brendle–Choi, 2017; Hoffman–Ilmanen–Martín–White, 2019; Sun–Zhihan Wang, 2023)38B2DD284E09
  2. 3.3 Rotational translating catenoids (Clutterbuck–Schnürer–Schulze, 2005; Hoffman–Ilmanen–Martín–White, 2019)FEA31DEDC853
  3. 3.4 Periodic translating tridents (X. H. Nguyen, 2009; Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)31505E08B028
  4. 3.5 Translators from grim-reaper configurations (X. H. Nguyen, 2010; X. H. Nguyen, 2012)E83F92A45A1F
  5. 3.6 Halldorsson helicoidal rotating translators0FA3E75DD419
  6. 3.7 Finite-genus translators with three ends (Dávila–del Pino–X. H. Nguyen, 2015; Hoffman–Ilmanen–Martín–White, 2019)8609A35E7601
  7. 3.8 Tilted grim-reaper cylinders9A8E5E528EEB
  8. 3.9 Strictly convex delta-wing translators9BBF93D61D7A
  9. 3.10 Translators with prescribed apex curvaturesDCD9F7224DF3
  10. 3.11 Scherkenoid translators (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)9E847D647A0C
  11. 3.12 The pitchfork translator (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019; Gama–Martín–Møller, 2022; Martín–Sáez–Tsiamis–White, 2024)64055D697644
  12. 3.13 Doubly periodic Scherk translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)73DEA085F2F9
  13. 3.14 Helicoid-like translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)0147E53C30CE
  14. 3.15 Annuloid translators in slabs (Gama–Martín–Møller, 2022; Hoffman–Martín–White, 2023)93A7D23F7FA3

1 family from lower dimensions →

ℝ⁴

  1. 3.16 Nonrotational entire convex translatorsBD00A99565D1
  2. 3.17 Joyce–Lee–Tsui small-angle Lagrangian translators16AF28E8CCDB
  3. 3.18 Explicit rotating Lagrangian translators5C21E6709CA8
  4. 3.19 Castro–Lerma Lagrangian translators from curve pairs14F0E7012E88
  5. 3.20 Convex slab translators with polyhedral asymptotics2F596CC616F1
  6. 3.21 Translators with an embedded rotational shrinking-torus end (Angenent, 1992; Sun–Zhihan Wang, 2023)C8173134CEA6
  7. 3.22 Translators with high-genus spherical-doubling ends (Kapouleas–McGrath, 2020; Sun–Zhihan Wang, 2023)A0A7AC0D200C
  8. 3.23 Translators with general closed-shrinker endsC44A1D57ECC3

5 families from lower dimensions →

Higher dimensions (N ≥ 5)

  1. 3.24 Su translators obtained by gluing grim cylinders96393C1B9F17
  2. 3.25 Su–Wood Lagrangian translating helicoids405ACD56C839
  3. 3.26 Su–Wood paired-profile translators31FE1060BFF7
  4. 3.27 Nonconvex entire graphical translators2E699B408ADA

10 families from lower dimensions →

Curved ambient spaces

  1. 3.28 Rotational translators in the hyperbolic-plane productE33234341F7D
  2. 3.29 Invariant translators in the Heisenberg group750539FAA8AF
  3. 3.30 Invariant translators in Sol geometry2BC769F3A4AD
  4. 3.31 Graphical translators over compact symmetric spaces (Fujii, 2021; Fujii–Koike, 2023)1102596C432B
  5. 3.32 Hyperbolic Killing translators and catenoids (de Lima–Ramos–dos Santos, 2023; de Lima–Ramos, 2026)EA9BCA366717
  6. 3.33 Helicoidal rotators-translators in the hyperbolic-plane product65C371A89575
  7. 3.34 Grim reapers in the hyperbolic-plane productFB29EFBDEAE7
  8. 3.35 Parabolic Killing grim reapers in hyperbolic space2C2BD4493223
  9. 3.36 Invariant translators in the special linear group9C3485CD7693
Chapter 4. ExpandersEE91DF9C2CB0

ℝ²

  1. 4.1 Expanding planar networks with a bounded cell2DAB3F0AC1D6
  2. 4.2 Self-expanding planar tree networks (Mazzeo–Saez, 2007; Ilmanen–Neves–Schulze, 2014)23B41DF2B95E
  3. 4.3 Expanding smoothings of planar corners (Halldórsson, 2010; Ding, 2015)D2DB1F7A1178

ℝ³

  1. 4.4 Graphical expanders from general Lipschitz cones152F8E9CDCFA
  2. 4.5 Two-sheeted expanders of rotational double conesADE41440A542
  3. 4.6 One-sheeted expanders of rotational double cones40FCBC562153
  4. 4.7 Rotational graphical expanders over round conesA178E87D4BA0
  5. 4.8 One-sided expanders from mean-convex cones5C7DD27422AD
  6. 4.9 Mountain-pass self-expanders665EA484B8DC
  7. 4.10 Three-ended expanders of positive genus (Shao–Zou, 2025; Hoffman–Martín–White, 2026)FC2EC5926764
  8. 4.11 Reported multiphase expanding surface constructionsBEE2570F7E0E
  9. 4.12 Knotted self-expanders38D2A75429F2

1 family from lower dimensions →

ℝ⁴

  1. 4.13 Lagrangian expander necks between transverse planes (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)6095D0C72380
  2. 4.14 Noncompact periodic Lagrangian expanders49998F6A2866
  3. 4.15 Two-convex Lagrangian graphical expanders5BCB359FC35B
  4. 4.16 Bi-rotational expanding hypersurfaces6BCE5569C7F9

6 families from lower dimensions →

Higher dimensions (N ≥ 5)

  1. 4.17 Lagrangian expanders with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)7681DAB7225B
  2. 4.18 Lee–Wang embedded quadric expanders3A7601A9FD31
  3. 4.19 Expanders based on generalized Lawson–Osserman maps258FD7148F89

10 families from lower dimensions →

Chapter 5. Other special flows52A98BA8F5C1

ℝ²

  1. 5.1 The ancient sine wave (Nakayama–Iizuka–Wadati, 1994; Angenent–You, 2018)D8ADD5B24C33
  2. 5.2 Purely rotating curve-shortening solitons (Halldórsson, 2010; Halldórsson, 2011; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)3A1BA479B53D
  3. 5.3 Rotating and expanding curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)E6DB29241FC5
  4. 5.4 Rotating and shrinking curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)3C52DD1F2A4E
  5. 5.5 The Angenent oval as an explicit ancient flow34E98635F82B
  6. 5.6 Eternal curves from chains of grim reapers (Angenent–You, 2018; Choi–Seo–Su–Zhao, 2026)321CCF32BF3F
  7. 5.7 Immersed ancient curves from grim-reaper chainsF7D04537B40F
  8. 5.8 Immersed ancient curves spanning a half-planeDCA2FD176411
  9. 5.9 Convex ancient free-boundary flows in balls (Bourni–Langford, 2021; Bourni–Langford, 2022)5E38E2470718
  10. 5.10 Compact nonconvex ancient spiral curvesF98988E964AC
  11. 5.11 Ancient curves with mixed boundary conditions4450DF902745
  12. 5.12 Ancient free-boundary curves in convex domains1D94C677A1B5

ℝ³

  1. 5.13 Minimal surfaces as stationary mean curvature flows (Chopp, 1993; Mantegazza, 2011)C9F0102B7272
  2. 5.14 Rotational degenerate neckpinch solutions1E4AD3904D39
  3. 5.15 Helicoidal minimal surfaces as stationary flows5BE3123E82F8
  4. 5.16 Eternal shrinking helices (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)281263153F94
  5. 5.17 Rotating and translating curve solitons9E45840987F2
  6. 5.18 Compact noncollapsed convex ancient ovals (Haslhofer–Hershkovits, 2013; Huisken–Sinestrari, 2014)12F33FD70B5C
  7. 5.19 Rotational collapsed ancient pancakes68F95608F072
  8. 5.20 Ancient flows from unstable minimal hypersurfaces (Melko, 2010; Mramor–Payne, 2019; Choi–Huang–Lee, 2024)C21B5E7431CB
  9. 5.21 The eternal reapernoid94DC745FE4C9
  10. 5.22 Non-self-similar ancient doughnuts1E48CF8C9E56
  11. 5.23 Ancient pancakes with polyhedral asymptoticsAD25A77F6BAA
  12. 5.24 Convex eternal flows that do not translateE62279A82D37
  13. 5.25 Nonplanar eternal curves from grim reapersAB3265DB1FB6
  14. 5.26 Non-self-similar flows emerging from conesEA5C07CB7D0F
  15. 5.27 Immortal annuli converging to a double plane00966387D6EF

4 families from lower dimensions →

ℝ⁴

  1. 5.28 Stationary special Lagrangian necks (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)380EC54D3A30
  2. 5.29 Ancient torus curves in higher codimension (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)B6082DA55C46
  3. 5.30 Ancient dense torus curves with irrational frequenciesE22AF7044670
  4. 5.31 Mean curvature evolution of the Whitney sphere2E15631C1593
  5. 5.32 Ancient isoparametric flows in Euclidean spaceF1B0B077B481
  6. 5.33 Ancient ovals with reduced symmetry37474A54C83A
  7. 5.34 Ancient stacked pancakes9C9E3EBC47D4

14 families from lower dimensions →

Higher dimensions (N ≥ 5)

  1. 5.35 Hypersurface singularities with bounded mean curvature88709443EE99
  2. 5.36 Lagrangian singularities with prescribed orbit modelsA3EE122FF71E
  3. 5.37 Su–Wood stationary products of Lawlor profilesC99B5440B24D
  4. 5.38 Stolarski–Su finite-time Lagrangian neck pinches71FD28B23406

22 families from lower dimensions →

Curved ambient spaces

  1. 5.39 Ancient isoparametric flows in spheres2DFF09123589
  2. 5.40 Eternal tori in perturbations of the round three-sphereEF033313C0BB
  3. 5.41 Isometric curve-shortening solitons in the hyperbolic plane6E7783E99D53
  4. 5.42 Finite-time Lagrangian neck pinches with circle symmetry06177EF7798F
  5. 5.43 Lagrangian torus necks collapsing at infinite time0AE8973A4338
  6. 5.44 Immortal compact flows converging to a double sphere0806DD487DC4
  7. 5.45 Ancient flows from unstable free-boundary minimal hypersurfaces1897F09E9330
  8. 5.46 Lagrangian sphere chains in Gibbons–Hawking spaces2AB1CBD37191
  9. 5.47 Immortal curve shortening with nonunique geodesic limits9C5BCCFA4320

1 family from lower dimensions →

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