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1. Introduction

• remarkRemark 1.1.1 (The source editions)912105D5B712Sources and conventions
• remarkRemark 1.1.2 (Mean curvature flow and normalization)4EDA6AEDEA07Sources and conventions
• remarkRemark 1.1.3 (Scope and illustrations)57833B199F67Sources and conventions

2. Shrinkers

• example2.1 Closed planar shrinking curves (Abresch–Langer, 1986; Halldórsson, 2010; Drugan–Lee–X. H. Nguyen, 2017; Colding–Minicozzi II, 2019)55D5743A6941Closed planar shrinking curves (Abresch–Langer, 1986; Halldórsson, 2010; Drugan–Lee–X. H. Nguyen, 2017; Colding–Minicozzi II, 2019)
• example2.2 Spoon-shaped shrinking network (Chen–Guo, 2007; Chang, 2021)DE792632B8AESpoon-shaped shrinking network (Chen–Guo, 2007; Chang, 2021)
• example2.3 Shrinking lens network (Chen–Guo, 2007; Chang, 2021)45920B126322Shrinking lens network (Chen–Guo, 2007; Chang, 2021)
• example2.4 Shrinking fish network (Chen–Guo, 2007; Chang, 2021)6E3C2D665CDDShrinking fish network (Chen–Guo, 2007; Chang, 2021)
• example2.5 Three-ray shrinking star (Chen–Guo, 2007; Chang, 2021)40F52B6CFB5CThree-ray shrinking star (Chen–Guo, 2007; Chang, 2021)
• example2.6 Four-ray shrinking star (Chen–Guo, 2007; Chang, 2021)010FE5696B4CFour-ray shrinking star (Chen–Guo, 2007; Chang, 2021)
• example2.7 Five-ray shrinking star (Chen–Guo, 2007; Chang, 2021)4C198F76FD40Five-ray shrinking star (Chen–Guo, 2007; Chang, 2021)
• example2.8 Shrinking rocket network (Chen–Guo, 2007; Chang, 2021)21D9CE8B056CShrinking rocket network (Chen–Guo, 2007; Chang, 2021)
• example2.9 Round shrinking spheres (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)5D6AFC4ECF1BRound shrinking spheres (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)
• example2.10 Planes through the shrinker center (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)97FD822B6630Planes through the shrinker center (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)
• example2.11 Nonclosed shrinking curves dense in an annulus4973C84D3B2BNonclosed shrinking curves dense in an annulus
• example2.12 Cisgeminate-eye shrinker8E8F9B32C891Cisgeminate-eye shrinker
• example2.13 Embedded rotational shrinking tori (Angenent, 1992; Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017; Berchenko-Kogan, 2018)10374A017921Embedded rotational shrinking tori (Angenent, 1992; Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017; Berchenko-Kogan, 2018)
• example2.14 Chopp cubical shrinker candidate04E9A0B8F725Chopp cubical shrinker candidate
• example2.15 Chopp octahedral shrinker candidate5D3FB6DFEE3FChopp octahedral shrinker candidate
• example2.16 Chopp punctured-torus saddle candidate3AA6C1836470Chopp punctured-torus saddle candidate
• example2.17 Chopp two-handled cone candidate48A9185FB30CChopp two-handled cone candidate
• example2.18 Numerical four-handled cone shrinker candidate (Chopp, 1994; Angenent–Ilmanen–Chopp, 1995)F576920EB05FNumerical four-handled cone shrinker candidate (Chopp, 1994; Angenent–Ilmanen–Chopp, 1995)
• example2.19 Generalized shrinking cylindersD20709465D01Generalized shrinking cylinders
• example2.20 Rotational shrinking trumpet ends41C51DD59791Rotational shrinking trumpet ends
• example2.21 Nguyen's one-ended shrinking saddlesB28BB2912147Nguyen's one-ended shrinking saddles
• example2.22 One-ended high-genus shrinkers from sphere–plane gluing (Kapouleas–Kleene–Møller, 2011; Sun–Zhichao Wang, 2018)A515DF7C7C60One-ended high-genus shrinkers from sphere–plane gluing (Kapouleas–Kleene–Møller, 2011; Sun–Zhichao Wang, 2018)
• example2.23 Closed shrinkers of large even genus via torus–sphere gluing (Colding–Minicozzi II, 2009; Møller, 2011; Drugan–Lee–X. H. Nguyen, 2017)A105C87CB202Closed shrinkers of large even genus via torus–sphere gluing (Colding–Minicozzi II, 2009; Møller, 2011; Drugan–Lee–X. H. Nguyen, 2017)
• example2.24 Immersed rotational shrinking spheres (Drugan, 2013; Drugan–Lee–X. H. Nguyen, 2017)FE38A115C52CImmersed rotational shrinking spheres (Drugan, 2013; Drugan–Lee–X. H. Nguyen, 2017)
• example2.25 Drugan–Kleene immersed shrinking familiesC28E16016422Drugan–Kleene immersed shrinking families
• example2.26 Shrinking ends asymptotic to isoparametric conesEDCE90129831Shrinking ends asymptotic to isoparametric cones
• example2.27 Ketover shrinkers with Platonic symmetryAE0CC89EB3E2Ketover shrinkers with Platonic symmetry
• example2.28 Variational embedded shrinking tori (Drugan–X. H. Nguyen, 2017; Chu–Sun, 2023)9F3C76F8391FVariational embedded shrinking tori (Drugan–X. H. Nguyen, 2017; Chu–Sun, 2023)
• example2.29 Lens-shaped shrinking surface clusters521D2D744A5FLens-shaped shrinking surface clusters
• example2.30 Closed shrinkers from spherical doubling826D61792601Closed shrinkers from spherical doubling
• example2.31 Sun–Wang–Zhou shrinkers of arbitrarily large entropy0177A2718D72Sun–Wang–Zhou shrinkers of arbitrarily large entropy
• example2.32 Noncompact shrinkers of every positive genus (Buzano–H. T. Nguyen–Schulz, 2021; Chu–Sun, 2023)40F2B3444C2ENoncompact shrinkers of every positive genus (Buzano–H. T. Nguyen–Schulz, 2021; Chu–Sun, 2023)
• example2.33 Chu–Sun genus-one shrinkers below the shrinking doughnutAE7721D6799FChu–Sun genus-one shrinkers below the shrinking doughnut
• example2.34 Ilmanen–White two-ended shrinkers4522A41972E4Ilmanen–White two-ended shrinkers
• example2.35 Ketover self-shrinking doubled planes9E9A8A1CC506Ketover self-shrinking doubled planes
• example2.36 Flow-generated Platonic and polygonal shrinkersA455BCCC51D0Flow-generated Platonic and polygonal shrinkers
• example2.37 Flow-generated sphere–plane shrinkers45B364E949D5Flow-generated sphere–plane shrinkers
• example2.38 Flow-generated three-ended shrinkers5CED6ED3B586Flow-generated three-ended shrinkers
• example2.39 Shao–Zou shrinkers with any prescribed number of ends4BFAC8529BEAShao–Zou shrinkers with any prescribed number of ends
• example2.40 Products of closed shrinking curvesAD717A30B415Products of closed shrinking curves
• example2.41 Higher-codimension spherical shrinkers (Smoczyk, 2005; Sun–Zhu, 2020)45597E4891F4Higher-codimension spherical shrinkers (Smoczyk, 2005; Sun–Zhu, 2020)
• example2.42 An embedded self-shrinking Möbius bundle (Lee–Mu-Tao Wang, 2007; Braxton–Lee–Zhu, 2026)CC2E5DD6BD78An embedded self-shrinking Möbius bundle (Lee–Mu-Tao Wang, 2007; Braxton–Lee–Zhu, 2026)
• example2.43 Compact periodic Lagrangian quadric shrinkers98593B70A49ECompact periodic Lagrangian quadric shrinkers
• example2.44 Noncompact periodic Lagrangian shrinkers2DED712E1192Noncompact periodic Lagrangian shrinkers
• example2.45 The Clifford shrinking torus in four-space635FBEF68A7EThe Clifford shrinking torus in four-space
• example2.46 Cylinders over embedded rotational shrinking tori (Angenent, 1992; Berchenko-Kogan, 2018; Chen–Lee–Sun–Zhao, 2026)FA65887C28F1Cylinders over embedded rotational shrinking tori (Angenent, 1992; Berchenko-Kogan, 2018; Chen–Lee–Sun–Zhao, 2026)
• example2.47 Park bi-rotational shrinking endsB9E60CF9DA60Park bi-rotational shrinking ends
• example2.48 Complete bi-rotational shrinking continuations313552CFDCE7Complete bi-rotational shrinking continuations
• example2.49 Lagrangian shrinkers with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)7077447C0B66Lagrangian shrinkers with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)
• example2.50 Lee–Wang embedded quadric shrinkers0EB941FB0D2BLee–Wang embedded quadric shrinkers
• example2.51 Riedler shrinkers from isoparametric foliations395E0B7648CARiedler shrinkers from isoparametric foliations
• example2.52 Yamamoto Lagrangian shrinkers in Calabi–Yau conesE540F08E41B0Yamamoto Lagrangian shrinkers in Calabi–Yau cones

3. Translators

• example3.1 The grim reaper and its cylinders (Hoffman–Ilmanen–Martín–White, 2018; Hoffman–Ilmanen–Martín–White, 2019)3B1939BC4D74The grim reaper and its cylinders (Hoffman–Ilmanen–Martín–White, 2018; Hoffman–Ilmanen–Martín–White, 2019)
• example3.2 The rotational bowl translator (Clutterbuck–Schnürer–Schulze, 2005; Brendle–Choi, 2017; Hoffman–Ilmanen–Martín–White, 2019; Sun–Zhihan Wang, 2023)ECF360A3D482The rotational bowl translator (Clutterbuck–Schnürer–Schulze, 2005; Brendle–Choi, 2017; Hoffman–Ilmanen–Martín–White, 2019; Sun–Zhihan Wang, 2023)
• example3.3 Rotational translating catenoids (Clutterbuck–Schnürer–Schulze, 2005; Hoffman–Ilmanen–Martín–White, 2019)72E3A9DEA98CRotational translating catenoids (Clutterbuck–Schnürer–Schulze, 2005; Hoffman–Ilmanen–Martín–White, 2019)
• example3.4 Periodic translating tridents (X. H. Nguyen, 2009; Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)27C982677562Periodic translating tridents (X. H. Nguyen, 2009; Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)
• example3.5 Translators from grim-reaper configurations (X. H. Nguyen, 2010; X. H. Nguyen, 2012)44161C7DCC84Translators from grim-reaper configurations (X. H. Nguyen, 2010; X. H. Nguyen, 2012)
• example3.6 Halldorsson helicoidal rotating translatorsE5C494420E4CHalldorsson helicoidal rotating translators
• example3.7 Finite-genus translators with three ends (Dávila–del Pino–X. H. Nguyen, 2015; Hoffman–Ilmanen–Martín–White, 2019)85D3C19BE7E6Finite-genus translators with three ends (Dávila–del Pino–X. H. Nguyen, 2015; Hoffman–Ilmanen–Martín–White, 2019)
• example3.8 Tilted grim-reaper cylinders56A3E33A9D04Tilted grim-reaper cylinders
• example3.9 Strictly convex delta-wing translatorsB72E7DC6B42EStrictly convex delta-wing translators
• example3.10 Translators with prescribed apex curvaturesD68656E88625Translators with prescribed apex curvatures
• example3.11 Scherkenoid translators (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)C6C5A44F27CCScherkenoid translators (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)
• example3.12 The pitchfork translator (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019; Gama–Martín–Møller, 2022; Martín–Sáez–Tsiamis–White, 2024)B5E7AB44607BThe pitchfork translator (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019; Gama–Martín–Møller, 2022; Martín–Sáez–Tsiamis–White, 2024)
• example3.13 Doubly periodic Scherk translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)D24DEEE97A5FDoubly periodic Scherk translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)
• example3.14 Helicoid-like translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)E6CC94303132Helicoid-like translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)
• example3.15 Annuloid translators in slabs (Gama–Martín–Møller, 2022; Hoffman–Martín–White, 2023)4887FC9B88DCAnnuloid translators in slabs (Gama–Martín–Møller, 2022; Hoffman–Martín–White, 2023)
• example3.16 Nonrotational entire convex translators0F00F9295CD9Nonrotational entire convex translators
• example3.17 Joyce–Lee–Tsui small-angle Lagrangian translatorsBCD2A9973070Joyce–Lee–Tsui small-angle Lagrangian translators
• example3.18 Explicit rotating Lagrangian translatorsA19CAED49B0DExplicit rotating Lagrangian translators
• example3.19 Castro–Lerma Lagrangian translators from curve pairsB4DA2A79EC96Castro–Lerma Lagrangian translators from curve pairs
• example3.20 Convex slab translators with polyhedral asymptoticsF05CE3FF1C12Convex slab translators with polyhedral asymptotics
• example3.21 Translators with an embedded rotational shrinking-torus end (Angenent, 1992; Sun–Zhihan Wang, 2023)BF18BDD64F8ETranslators with an embedded rotational shrinking-torus end (Angenent, 1992; Sun–Zhihan Wang, 2023)
• example3.22 Translators with high-genus spherical-doubling ends (Kapouleas–McGrath, 2020; Sun–Zhihan Wang, 2023)9B1B88A4317FTranslators with high-genus spherical-doubling ends (Kapouleas–McGrath, 2020; Sun–Zhihan Wang, 2023)
• example3.23 Translators with general closed-shrinker ends191836516CC9Translators with general closed-shrinker ends
• example3.24 Su translators obtained by gluing grim cylinders164F129D254DSu translators obtained by gluing grim cylinders
• example3.25 Su–Wood Lagrangian translating helicoids6386C95CB0EASu–Wood Lagrangian translating helicoids
• example3.26 Su–Wood paired-profile translatorsFDCF3D0F683ESu–Wood paired-profile translators
• example3.27 Nonconvex entire graphical translators879611BCBD19Nonconvex entire graphical translators
• example3.28 Rotational translators in the hyperbolic-plane product86412C87CEBERotational translators in the hyperbolic-plane product
• example3.29 Invariant translators in the Heisenberg group98084CBA4530Invariant translators in the Heisenberg group
• example3.30 Invariant translators in Sol geometry96A9DA4F04BFInvariant translators in Sol geometry
• example3.31 Graphical translators over compact symmetric spaces (Fujii, 2021; Fujii–Koike, 2023)B6DC0DA73EFFGraphical translators over compact symmetric spaces (Fujii, 2021; Fujii–Koike, 2023)
• example3.32 Hyperbolic Killing translators and catenoids (de Lima–Ramos–dos Santos, 2023; de Lima–Ramos, 2026)C13AFD802566Hyperbolic Killing translators and catenoids (de Lima–Ramos–dos Santos, 2023; de Lima–Ramos, 2026)
• example3.33 Helicoidal rotators-translators in the hyperbolic-plane product32D1DDDBF12BHelicoidal rotators-translators in the hyperbolic-plane product
• example3.34 Grim reapers in the hyperbolic-plane product38A33FD9E29CGrim reapers in the hyperbolic-plane product
• example3.35 Parabolic Killing grim reapers in hyperbolic space69BB4AD6B5E2Parabolic Killing grim reapers in hyperbolic space
• example3.36 Invariant translators in the special linear group3535C925421EInvariant translators in the special linear group

4. Expanders

• example4.1 Expanding planar networks with a bounded cell4D33C840252CExpanding planar networks with a bounded cell
• example4.2 Self-expanding planar tree networks (Mazzeo–Saez, 2007; Ilmanen–Neves–Schulze, 2014)16898F334139Self-expanding planar tree networks (Mazzeo–Saez, 2007; Ilmanen–Neves–Schulze, 2014)
• example4.3 Expanding smoothings of planar corners (Halldórsson, 2010; Ding, 2015)72808FA576A9Expanding smoothings of planar corners (Halldórsson, 2010; Ding, 2015)
• example4.4 Graphical expanders from general Lipschitz cones9A84A12D805CGraphical expanders from general Lipschitz cones
• example4.5 Two-sheeted expanders of rotational double cones4F35D93CA4BDTwo-sheeted expanders of rotational double cones
• example4.6 One-sheeted expanders of rotational double cones3A4775A9BD2COne-sheeted expanders of rotational double cones
• example4.7 Rotational graphical expanders over round conesD3CC5E948716Rotational graphical expanders over round cones
• example4.8 One-sided expanders from mean-convex conesA6044506F79AOne-sided expanders from mean-convex cones
• example4.9 Mountain-pass self-expanders1EA720390291Mountain-pass self-expanders
• example4.10 Three-ended expanders of positive genus (Shao–Zou, 2025; Hoffman–Martín–White, 2026)3A8C3B21B572Three-ended expanders of positive genus (Shao–Zou, 2025; Hoffman–Martín–White, 2026)
• example4.11 Reported multiphase expanding surface constructions55F8A309578DReported multiphase expanding surface constructions
• example4.12 Knotted self-expandersDA3FCE2F354FKnotted self-expanders
• example4.13 Lagrangian expander necks between transverse planes (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)E25A2DDDEA95Lagrangian expander necks between transverse planes (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)
• example4.14 Noncompact periodic Lagrangian expanders3823D6B50DAFNoncompact periodic Lagrangian expanders
• example4.15 Two-convex Lagrangian graphical expanders9910637B58B3Two-convex Lagrangian graphical expanders
• example4.16 Bi-rotational expanding hypersurfaces060795635061Bi-rotational expanding hypersurfaces
• example4.17 Lagrangian expanders with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)108077B0C2DELagrangian expanders with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)
• example4.18 Lee–Wang embedded quadric expandersA1A5802C8577Lee–Wang embedded quadric expanders
• example4.19 Expanders based on generalized Lawson–Osserman maps42620E4B40CFExpanders based on generalized Lawson–Osserman maps

5. Other special flows

• example5.1 The ancient sine wave (Nakayama–Iizuka–Wadati, 1994; Angenent–You, 2018)3BF94583DFB1The ancient sine wave (Nakayama–Iizuka–Wadati, 1994; Angenent–You, 2018)
• example5.2 Purely rotating curve-shortening solitons (Halldórsson, 2010; Halldórsson, 2011; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)999F58954ACBPurely rotating curve-shortening solitons (Halldórsson, 2010; Halldórsson, 2011; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)
• example5.3 Rotating and expanding curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)23FC29A5131CRotating and expanding curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)
• example5.4 Rotating and shrinking curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)EF429B51A962Rotating and shrinking curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)
• example5.5 The Angenent oval as an explicit ancient flowB008243C5D96The Angenent oval as an explicit ancient flow
• example5.6 Eternal curves from chains of grim reapers (Angenent–You, 2018; Choi–Seo–Su–Zhao, 2026)B7D6B1E7217FEternal curves from chains of grim reapers (Angenent–You, 2018; Choi–Seo–Su–Zhao, 2026)
• example5.7 Immersed ancient curves from grim-reaper chains104CF0DAFD60Immersed ancient curves from grim-reaper chains
• example5.8 Immersed ancient curves spanning a half-plane4B3B50761BB9Immersed ancient curves spanning a half-plane
• example5.9 Convex ancient free-boundary flows in balls (Bourni–Langford, 2021; Bourni–Langford, 2022)2CB973E94410Convex ancient free-boundary flows in balls (Bourni–Langford, 2021; Bourni–Langford, 2022)
• example5.10 Compact nonconvex ancient spiral curves505A2B69BD4FCompact nonconvex ancient spiral curves
• example5.11 Ancient curves with mixed boundary conditionsFB8BC9B105B9Ancient curves with mixed boundary conditions
• example5.12 Ancient free-boundary curves in convex domains9FAA74B5E7DAAncient free-boundary curves in convex domains
• example5.13 Minimal surfaces as stationary mean curvature flows (Chopp, 1993; Mantegazza, 2011)AC1C0D984A58Minimal surfaces as stationary mean curvature flows (Chopp, 1993; Mantegazza, 2011)
• example5.14 Rotational degenerate neckpinch solutions922034E946EBRotational degenerate neckpinch solutions
• example5.15 Helicoidal minimal surfaces as stationary flowsE9086532F53AHelicoidal minimal surfaces as stationary flows
• example5.16 Eternal shrinking helices (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)1EF23C4EFFC0Eternal shrinking helices (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)
• example5.17 Rotating and translating curve solitons955BE4106D88Rotating and translating curve solitons
• example5.18 Compact noncollapsed convex ancient ovals (Haslhofer–Hershkovits, 2013; Huisken–Sinestrari, 2014)56D296501918Compact noncollapsed convex ancient ovals (Haslhofer–Hershkovits, 2013; Huisken–Sinestrari, 2014)
• example5.19 Rotational collapsed ancient pancakes325F4A3220BCRotational collapsed ancient pancakes
• example5.20 Ancient flows from unstable minimal hypersurfaces (Melko, 2010; Mramor–Payne, 2019; Choi–Huang–Lee, 2024)8A13A6308368Ancient flows from unstable minimal hypersurfaces (Melko, 2010; Mramor–Payne, 2019; Choi–Huang–Lee, 2024)
• example5.21 The eternal reapernoid8EACFCA7AE0FThe eternal reapernoid
• example5.22 Non-self-similar ancient doughnuts7A6AD2F58C6ENon-self-similar ancient doughnuts
• example5.23 Ancient pancakes with polyhedral asymptotics8C2928E390C8Ancient pancakes with polyhedral asymptotics
• example5.24 Convex eternal flows that do not translateFA0C7583A161Convex eternal flows that do not translate
• example5.25 Nonplanar eternal curves from grim reapers1A07DD8D8A69Nonplanar eternal curves from grim reapers
• example5.26 Non-self-similar flows emerging from conesAD7B9EFD3360Non-self-similar flows emerging from cones
• example5.27 Immortal annuli converging to a double planeA50018836027Immortal annuli converging to a double plane
• example5.28 Stationary special Lagrangian necks (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)263868E061E7Stationary special Lagrangian necks (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)
• example5.29 Ancient torus curves in higher codimension (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)D28FAF32B556Ancient torus curves in higher codimension (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)
• example5.30 Ancient dense torus curves with irrational frequencies3E693532AA77Ancient dense torus curves with irrational frequencies
• example5.31 Mean curvature evolution of the Whitney sphere245DFF2BED27Mean curvature evolution of the Whitney sphere
• example5.32 Ancient isoparametric flows in Euclidean space199B3435F93BAncient isoparametric flows in Euclidean space
• example5.33 Ancient ovals with reduced symmetryF18DC5F92665Ancient ovals with reduced symmetry
• example5.34 Ancient stacked pancakes1A2B6DF32B79Ancient stacked pancakes
• example5.35 Hypersurface singularities with bounded mean curvatureD9D28F3C8E4FHypersurface singularities with bounded mean curvature
• example5.36 Lagrangian singularities with prescribed orbit models5DA3E72E4195Lagrangian singularities with prescribed orbit models
• example5.37 Su–Wood stationary products of Lawlor profiles3CC118326D79Su–Wood stationary products of Lawlor profiles
• example5.38 Stolarski–Su finite-time Lagrangian neck pinches41748F7ED1C1Stolarski–Su finite-time Lagrangian neck pinches
• example5.39 Ancient isoparametric flows in spheres51259B78DC80Ancient isoparametric flows in spheres
• example5.40 Eternal tori in perturbations of the round three-sphere4F2C4D7007EDEternal tori in perturbations of the round three-sphere
• example5.41 Isometric curve-shortening solitons in the hyperbolic plane39A0C4D2B77AIsometric curve-shortening solitons in the hyperbolic plane
• example5.42 Finite-time Lagrangian neck pinches with circle symmetryA25A9D4CCA2BFinite-time Lagrangian neck pinches with circle symmetry
• example5.43 Lagrangian torus necks collapsing at infinite time2DFF070BC43FLagrangian torus necks collapsing at infinite time
• example5.44 Immortal compact flows converging to a double sphere225751B3C7E8Immortal compact flows converging to a double sphere
• example5.45 Ancient flows from unstable free-boundary minimal hypersurfaces3846578EAD16Ancient flows from unstable free-boundary minimal hypersurfaces
• example5.46 Lagrangian sphere chains in Gibbons–Hawking spacesDE8829990703Lagrangian sphere chains in Gibbons–Hawking spaces
• example5.47 Immortal curve shortening with nonunique geodesic limitsCC2635F97DC0Immortal curve shortening with nonunique geodesic limits