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Calibration energy and mean curvature flow

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Pith's one-line read The calibration energy yields an exact dissipation identity along proper mean curvature flows, forcing every proper self-expander with finite constant-coefficient calibration energy to be a plane in all dimensions and codimensions.

desk verdict The calibration energy and its exact dissipation identity along MCF are the real novelty, yielding plane rigidity for finite-energy self-expanders, but the local-volume bound must be shown to hold under finite energy or the applications have a gap. read the letter →

arxiv 2606.04586 v2 pith:KZA4RVCA submitted 2026-06-03 math.DG math.AP

Tatsuya Miura , Fabian Rupp This is my paper · ORCID
classification math.DGmath.AP
keywords calibrationenergymeancurvatureflowself-expandersrigiditysolitonsdissipationidentityorientedimmersions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper defines a calibration energy for oriented immersions into Euclidean space that quantifies deviation from calibrated geometry and remains finite for some infinite-volume immersions. A null-Lagrangian structure ensures the energy shares the same first variation as the volume functional. The central result is an exact dissipation identity for this energy along oriented proper mean curvature flows, which holds under only a mild local-volume bound. This identity supplies a new finite variational setting for the flow and directly implies rigidity: every proper self-expander with finite constant-coefficient calibration energy must be a plane.

What carries the argument

The calibration energy, a functional on oriented immersions that measures deviation from calibrated geometry, shares the first variation of volume because of its null-Lagrangian structure, and dissipates monotonically along mean curvature flow.

What would settle it

A single non-planar proper self-expander in Euclidean space carrying finite constant-coefficient calibration energy would falsify the rigidity statement.

Watch

Extended reading notes

Core claim

Every proper self-expander with finite constant-coefficient calibration energy must be a plane in all dimensions and codimensions. The proof rests on establishing an exact dissipation identity for the calibration energy along oriented proper mean curvature flows under a mild local-volume bound; the same identity also yields convergence statements for two-dimensional immortal solutions.

Load-bearing premise

A mild local-volume bound is required for the dissipation identity to hold along the flow.

Editorial extensions

If this is right

  • Proper self-expanders are rigid: only planes admit finite calibration energy.
  • Two-dimensional immortal mean curvature flows converge when the calibration energy is finite.
  • The energy supplies a variational framework for mean curvature flow that works even when total volume is infinite.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The dissipation identity might be adapted to study mean curvature flow in settings where volume is infinite but another controlled quantity replaces the local-volume bound.
  • Similar energy constructions could be explored for other curvature flows or for immersions into manifolds with calibrated geometries.
  • The rigidity result suggests that finite calibration energy may serve as a compactness criterion for sequences of self-expanders.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper introduces the calibration energy for oriented immersions into Euclidean space, which quantifies deviation from calibrated geometry while remaining finite for some infinite-volume immersions. A null-Lagrangian structure ensures the energy has the same first variation as the volume functional. The central result is an exact dissipation identity for the calibration energy along oriented proper mean curvature flows in arbitrary dimensions and codimensions, proved under a mild local-volume bound. This framework is applied to obtain rigidity: every proper self-expander with finite constant-coefficient calibration energy must be a plane, along with convergence results for two-dimensional immortal solutions.

Significance. If the dissipation identity and its application to self-expanders hold, the work supplies a new finite-energy variational setting for mean curvature flow that extends beyond the finite-volume case. The exact (non-approximate) dissipation identity, the parameter-free character of the energy definition, and the all-dimensions/codimensions rigidity statement for self-expanders are concrete strengths. The manuscript introduces a genuinely new energy functional rather than recycling an existing one.

major comments (1)
  1. [Abstract / applications section] Abstract and the section deriving the rigidity theorem for self-expanders: the exact dissipation identity is established only under the mild local-volume bound, yet the claim that every proper self-expander with finite calibration energy is a plane applies the identity without separately verifying that finite energy plus properness implies the bound (or that the bound holds automatically for this class). If the bound can fail while energy remains finite, the dissipation step does not apply and the rigidity conclusion does not follow.
minor comments (1)
  1. [Introduction] Notation for the constant-coefficient calibration energy should be introduced with an explicit formula in the introduction rather than deferred.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment of the paper's contributions. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract / applications section] Abstract and the section deriving the rigidity theorem for self-expanders: the exact dissipation identity is established only under the mild local-volume bound, yet the claim that every proper self-expander with finite calibration energy is a plane applies the identity without separately verifying that finite energy plus properness implies the bound (or that the bound holds automatically for this class). If the bound can fail while energy remains finite, the dissipation step does not apply and the rigidity conclusion does not follow.

    Authors: We agree that the manuscript should explicitly verify that the mild local-volume bound holds for the class of proper self-expanders with finite calibration energy, so that the dissipation identity applies without additional assumptions. In the revised version we will insert a short lemma establishing this implication (using the self-expander equation together with the finiteness of the calibration energy to control local volume growth). This will make the passage from the dissipation identity to the rigidity statement fully rigorous. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: new energy functional with independent dissipation identity under explicit hypothesis

full rationale

The calibration energy is introduced as a new quantity quantifying deviation from calibrated geometry, with its null-Lagrangian structure and first-variation equivalence to volume stated as direct properties of the definition. The exact dissipation identity is derived along proper MCF under the separately stated mild local-volume bound; the rigidity result for self-expanders is obtained by applying this identity once the bound is assumed. No step reduces a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames a known result; the chain remains self-contained against the stated assumptions.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The claims rest on standard differential geometry of immersions and the newly introduced energy; no free parameters or invented entities are visible from the abstract.

assumptions (1)
  • standard math Oriented proper immersions into Euclidean space admit a well-defined mean curvature flow under the stated local volume bound.
    Invoked to set up the flow and the identity.
invented entities (1)
  • Calibration energy
    purpose: Quantifies deviation from calibrated geometry while sharing first variation with volume.
    Newly defined functional central to the dissipation identity.

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Cite this review

Pith. "Pith review of Calibration energy and mean curvature flow." pith.science (2026). https://pith.science/paper/KZA4RVCA

@misc{pith2026260604586,
  author       = {Pith},
  title        = {Pith review of: Calibration energy and mean curvature flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZA4RVCA}},
  note         = {Machine review of arXiv:2606.04586}
}
read the original abstract

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along oriented, proper mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. This provides a new, finite variational framework for mean curvature flow beyond the finite-volume setting. Our result yields several applications, including rigidity for solitons and convergence for two-dimensional immortal solutions. In particular, every proper self-expander with finite constant-coefficient calibration energy must be a plane in all dimensions and codimensions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

89 extracted references · 6 canonical work pages

  1. [1]

    Angenent, T

    S. Angenent, T. Ilmanen, and D. L. Chopp,A computed example of nonuniqueness of mean curvature flow inR 3, Comm. Partial Differential Equations20(1995), no. 11-12, 1937–1958. MR1361726

    1995

  2. [2]

    Bernstein,Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends, Amer

    J. Bernstein,Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends, Amer. J. Math.142(2020), no. 6, 1897–1929. MR4176548

    2020

  3. [3]

    Bernstein and L

    J. Bernstein and L. Wang,A mountain-pass theorem for asymptotically conical self- expanders, Peking Math. J.5(2022), no. 2, 213–278. MR4492654

    2022

  4. [4]

    Bernstein and L

    J. Bernstein and L. Wang,Topological uniqueness for self-expanders of small entropy, Camb. J. Math.10(2022), no. 4, 785–833. MR4524828

    2022

  5. [5]

    Bochnak, M

    J. Bochnak, M. Coste, and M.-F. Roy,Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer- Verlag, Berlin, 1998. Translated from the 1987 French original, Revised by the authors. MR1659509

    1998

  6. [6]

    Bombieri, E

    E. Bombieri, E. De Giorgi, and E. Giusti,Minimal cones and the Bernstein problem, Invent. Math.7(1969), 243–268. MR250205

    1969

  7. [7]

    K. A. Brakke,The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, NJ, 1978. MR485012

    1978

  8. [8]

    Brendle and G

    S. Brendle and G. Huisken,Mean curvature flow with surgery of mean convex surfaces in R3, Invent. Math.203(2016), no. 2, 615–654. MR3455158

    2016

Show all 89 references
  1. [9]

    Cannas da Silva,Lectures on symplectic geometry, Lecture Notes in Mathematics, vol

    A. Cannas da Silva,Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR1853077

    2001

  2. [10]

    Cao and H

    H.-D. Cao and H. Li,A gap theorem for self-shrinkers of the mean curvature flow in ar- bitrary codimension, Calc. Var. Partial Differential Equations46(2013), no. 3-4, 879–889. MR3018176

    2013

  3. [11]

    Chen and L

    B.-L. Chen and L. Yin,Uniqueness and pseudolocality theorems of the mean curvature flow, Comm. Anal. Geom.15(2007), no. 3, 435–490. MR2379801

    2007

  4. [12]

    Cheng,On the Type IIb solutions to mean curvature flow, J

    L. Cheng,On the Type IIb solutions to mean curvature flow, J. Differential Equations269 (2020), no. 10, 8350–8369. MR4113204

    2020

  5. [13]

    Cheng and N

    L. Cheng and N. Sesum,Asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces, Comm. Anal. Geom.26(2018), no. 5, 1079–1101. MR3900480

    2018

  6. [14]

    Cheng and Y

    L. Cheng and Y. Zhang,No breather theorems for the mean curvature flow, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)27(2026), no. 1, 145–165. MR5070682

    2026

  7. [15]

    S. S. Chern and J. G. Wolfson,Minimal surfaces by moving frames, Amer. J. Math.105 (1983), no. 1, 59–83. MR692106

    1983

  8. [16]

    Clutterbuck, O

    J. Clutterbuck, O. C. Schn¨ urer, and F. Schulze,Stability of translating solutions to mean cur- vature flow, Calc. Var. Partial Differential Equations29(2007), no. 3, 281–293. MR2321890

    2007

  9. [17]

    T. H. Colding and W. P. Minicozzi II,Generic mean curvature flow I: generic singularities, Ann. of Math. (2)175(2012), no. 2, 755–833. MR2993752

    2012

  10. [18]

    Dadok, R

    J. Dadok, R. Harvey, and F. Morgan,Calibrations onR 8, Trans. Amer. Math. Soc.307 (1988), no. 1, 1–40. MR936802

    1988

  11. [19]

    Dadok and F

    J. Dadok and F. R. Harvey,Calibrations and spinors, Acta Math.170(1993), no. 1, 83–120. MR1208563

    1993

  12. [20]

    Dadok and R

    J. Dadok and R. Harvey,Calibrations onR 6, Duke Math. J.50(1983), no. 4, 1231–1243. MR726326

    1983

  13. [21]

    Daskalopoulos and M

    P. Daskalopoulos and M. Saez,Uniqueness of entire graphs evolving by mean curvature flow, J. Reine Angew. Math.796(2023), 201–227. MR4554469

    2023

  14. [22]

    del Pino, M

    M. del Pino, M. Kowalczyk, and J. Wei,On De Giorgi’s conjecture in dimensionN≥9, Ann. of Math. (2)174(2011), no. 3, 1485–1569. MR2846486

    2011

  15. [23]

    Deruelle and F

    A. Deruelle and F. Schulze,Generic uniqueness of expanders with vanishing relative entropy, Math. Ann.377(2020), no. 3-4, 1095–1127. MR4126890

    2020

  16. [24]

    Ding,Minimal cones and self-expanding solutions for mean curvature flows, Math

    Q. Ding,Minimal cones and self-expanding solutions for mean curvature flows, Math. Ann. 376(2020), no. 1-2, 359–405. MR4055164 CALIBRATION ENERGY AND MEAN CURVATURE FLOW 41

    2020

  17. [25]

    H. T. Dung, N. T. Dung, and T. Q. Huy,Rigidity and vanishing theorems for complete translating solitons, Manuscripta Math.172(2023), no. 1-2, 331–352. MR4632930

    2023

  18. [26]

    Ecker,Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equa- tions and their Applications, vol

    K. Ecker,Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equa- tions and their Applications, vol. 57, Birkh¨ auser Boston, Inc., Boston, MA, 2004. MR2024995

    2004

  19. [27]

    Ecker and G

    K. Ecker and G. Huisken,Mean curvature evolution of entire graphs, Ann. of Math. (2)130 (1989), no. 3, 453–471. MR1025164

    1989

  20. [28]

    Ecker and G

    K. Ecker and G. Huisken,Interior estimates for hypersurfaces moving by mean curvature, Invent. Math.105(1991), no. 3, 547–569. MR1117150

    1991

  21. [29]

    L. C. Evans,Partial differential equations, Second, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR2597943

    2010

  22. [30]

    Fischer, S

    J. Fischer, S. Hensel, T. Laux, and T. M. Simon,The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, J. Eur. Math. Soc. (2025). published online first

    2025

  23. [31]

    Fu,An analogue of Bernstein’s theorem, Houston J

    L. Fu,An analogue of Bernstein’s theorem, Houston J. Math.24(1998), no. 3, 415–419. MR1686614

    1998

  24. [32]

    E. S. Gama,Translating solitonsC 1-asymptotic to two half-hyperplanes, Proc. Amer. Math. Soc.148(2020), no. 2, 851–858. MR4052220

    2020

  25. [33]

    E. S. Gama and F. Mart´ ın,Translating solitons of the mean curvature flow asymptotic to hyperplanes inR n+1, Int. Math. Res. Not. IMRN24(2020), 10114–10153. MR4190399

    2020

  26. [34]

    Guo,Asymptotic behavior and stability of mean curvature flow with a conical end, Adv

    S.-H. Guo,Asymptotic behavior and stability of mean curvature flow with a conical end, Adv. Math.375(2020), 107408, 63. MR4170224

    2020

  27. [35]

    F. R. Harvey,Spinors and calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Inc., Boston, MA, 1990. MR1045637

    1990

  28. [36]

    Harvey and H

    R. Harvey and H. B. Lawson Jr.,Calibrated geometries, Acta Math.148(1982), 47–157. MR666108

    1982

  29. [37]

    Harvey and F

    R. Harvey and F. Morgan,The faces of the Grassmannian of three-planes inR 7 (calibrated geometries onR 7), Invent. Math.83(1986), no. 2, 191–228. MR818350

    1986

  30. [38]

    Helmensdorfer,A model for the behavior of fluid droplets based on mean curvature flow, SIAM J

    S. Helmensdorfer,A model for the behavior of fluid droplets based on mean curvature flow, SIAM J. Math. Anal.44(2012), no. 3, 1359–1371. MR2982716

    2012

  31. [39]

    Hershkovits,Translators asymptotic to cylinders, J

    O. Hershkovits,Translators asymptotic to cylinders, J. Reine Angew. Math.766(2020), 61–

    2020

  32. [40]

    Hoffman, T

    D. Hoffman, T. Ilmanen, F. Mart´ ın, and B. White,Notes on translating solitons for mean cur- vature flow, Minimal surfaces: integrable systems and visualisation, [2021]©2021, pp. 147–

    2021

  33. [41]

    Hoffman, F

    D. Hoffman, F. Mart´ ın, and B. White,Translating annuli for mean curvature flow, Adv. Math.455(2024), Paper No. 109875, 109. MR4784721

    2024

  34. [42]

    Huisken,Asymptotic behavior for singularities of the mean curvature flow, J

    G. Huisken,Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom.31(1990), no. 1, 285–299. MR1030675

    1990

  35. [43]

    Ilmanen,Lectures on mean curvature flow and related equations, arXiv preprint arXiv:2601.21952v1 (1995)

    T. Ilmanen,Lectures on mean curvature flow and related equations, arXiv preprint arXiv:2601.21952v1 (1995)

    1995

  36. [44]

    Ilmanen, A

    T. Ilmanen, A. Neves, and F. Schulze,On short time existence for the planar network flow, J. Differential Geom.111(2019), no. 1, 39–89. MR3909904

    2019

  37. [45]

    Isenberg, H

    J. Isenberg, H. Wu, and Z. Zhang,On the precise asymptotics of Type-IIb solutions to mean curvature flow, Trans. Amer. Math. Soc. Ser. B9(2022), 564–585. MR4442627

    2022

  38. [46]

    D. D. Joyce,Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, vol. 12, Oxford University Press, Oxford, 2007. MR2292510

    2007

  39. [47]

    Khan,The structure of translating surfaces with finite total curvature, Calc

    I. Khan,The structure of translating surfaces with finite total curvature, Calc. Var. Partial Differential Equations62(2023), no. 3, Paper No. 104, 28. MR4553955

    2023

  40. [48]

    Khan,Uniqueness of asymptotically conical higher codimension self-shrinkers and self- expanders, J

    I. Khan,Uniqueness of asymptotically conical higher codimension self-shrinkers and self- expanders, J. Reine Angew. Math.805(2023), 55–99. MR4669035

    2023

  41. [49]

    Kunikawa,Bernstein-type theorem of translating solitons in arbitrary codimension with flat normal bundle, Calc

    K. Kunikawa,Bernstein-type theorem of translating solitons in arbitrary codimension with flat normal bundle, Calc. Var. Partial Differential Equations54(2015), no. 2, 1331–1344. MR3396413

    2015

  42. [50]

    Kuwert and R

    E. Kuwert and R. Sch¨ atzle,Gradient flow for the Willmore functional, Comm. Anal. Geom. 10(2002), no. 2, 307–339. MR1900754

    2002

  43. [51]

    Kuwert and R

    E. Kuwert and R. Sch¨ atzle,Removability of point singularities of Willmore surfaces, Ann. of Math. (2)160(2004), no. 1, 315–357. MR2119722 42 T. MIURA AND F. RUPP

    2004

  44. [52]

    N. Q. Le and N. Sesum,Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers, Comm. Anal. Geom.19(2011), no. 4, 633–659. MR2880211

    2011

  45. [53]

    Li,Non-polynomial entire solutions to Hessian equations, Calc

    C. Li,Non-polynomial entire solutions to Hessian equations, Calc. Var. Partial Differential Equations60(2021), no. 4, Paper No. 123, 6. MR4277327

    2021

  46. [54]

    Lynch and G

    S. Lynch and G. Tinaglia,Translators Asymptotic to Planes, J. Geom. Anal.36(2026), no. 1, Paper No. 27. MR4996046

    2026

  47. [55]

    J. M. S. Ma,Parabolic Omori-Yau maximum principle for mean curvature flow and some applications, J. Geom. Anal.28(2018), no. 4, 3183–3195. MR3881969

    2018

  48. [56]

    Magni and C

    A. Magni and C. Mantegazza,Some remarks on Huisken’s monotonicity formula for mean curvature flow, Singularities in nonlinear evolution phenomena and applications, 2009, pp. 157–169. MR2528703

    2009

  49. [57]

    Mart´ ın, J

    F. Mart´ ın, J. P´ erez-Garc´ ıa, A. Savas-Halilaj, and K. Smoczyk,A characterization of the grim reaper cylinder, J. Reine Angew. Math.746(2019), 209–234. MR3895630

    2019

  50. [58]

    Mart´ ın, A

    F. Mart´ ın, A. Savas-Halilaj, and K. Smoczyk,On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations54(2015), no. 3, 2853–2882. MR3412395

    2015

  51. [59]

    McDuff and D

    D. McDuff and D. Salamon,J-holomorphic curves and symplectic topology, Second, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2012. MR2954391

    2012

  52. [60]

    J. H. Michael and L. M. Simon,Sobolev and mean-value inequalities on generalized subman- ifolds ofR n, Comm. Pure Appl. Math.26(1973), 361–379. MR344978

    1973

  53. [61]

    Miura,Elastic curves and phase transitions, Math

    T. Miura,Elastic curves and phase transitions, Math. Ann.376(2020), no. 3-4, 1629–1674. MR4081125

    2020

  54. [62]

    Miura, M

    T. Miura, M. M¨ uller, and F. Rupp,Optimal thresholds for preserving embeddedness of elastic flows, Amer. J. Math.147(2025), no. 1, 33–80. MR4861585

    2025

  55. [63]

    Miura and F

    T. Miura and F. Rupp,Embeddedness and graphicality of the elastic flow for complete curves, arXiv preprint arXiv:2508.18979v1 (2025)

    2025

  56. [64]

    Miura and F

    T. Miura and F. Rupp,A new energy method for shortening and straightening complete curves, J. Reine Angew. Math.830(2026), 1–50. MR5008349

    2026

  57. [65]

    Miura and G

    T. Miura and G. Wheeler,Uniqueness and minimality of Euler’s elastica with monotone curvature, arXiv:2402.12771, to appear in J. Eur. Math. Soc. (JEMS) (2024)

    2024

  58. [66]

    M¨ uller and V.ˇSver´ ak,On surfaces of finite total curvature, J

    S. M¨ uller and V.ˇSver´ ak,On surfaces of finite total curvature, J. Differential Geom.42(1995), no. 2, 229–258. MR1366547

    1995

  59. [67]

    Neves,Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent

    A. Neves,Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math.168(2007), no. 3, 449–484. MR2299559

    2007

  60. [68]

    F. Otto, R. Schubert, and M. G. Westdickenberg,Convergence to the planar interface for a nonlocal free-boundary evolution, Comm. Pure Appl. Math.78(2025), no. 1, 161–208. MR4822909

    2025

  61. [69]

    Perelman,The entropy formula for the Ricci flow and its geometric applications, arXiv preprint math (2002)

    G. Perelman,The entropy formula for the Ricci flow and its geometric applications, arXiv preprint math (2002)

    2002

  62. [70]

    Qiu,A Bernstein type result of translating solitons, Calc

    H. Qiu,A Bernstein type result of translating solitons, Calc. Var. Partial Differential Equa- tions61(2022), no. 6, Paper No. 228, 9. MR4496501

    2022

  63. [71]

    Qiu,Bernstein type theorems of translating solitons of the mean curvature flow in higher codimension, Calc

    H. Qiu,Bernstein type theorems of translating solitons of the mean curvature flow in higher codimension, Calc. Var. Partial Differential Equations63(2024), no. 2, Paper No. 51, 12. MR4701858

    2024

  64. [72]

    Savas Halilaj and K

    A. Savas Halilaj and K. Smoczyk,Codimension two mean curvature flow of entire graphs, J. Lond. Math. Soc. (2)110(2024), no. 5, Paper No. e13000, 33. MR4810572

    2024

  65. [73]

    Shao and J

    G. Shao and J. Zou,Self-expanders of positive genus, arXiv preprint arXiv:2507.19428v2 (2025)

    2025

  66. [74]

    Smoczyk,A canonical way to deform a Lagrangian submanifold, arXiv preprint arXiv:dg- ga (1996)

    K. Smoczyk,A canonical way to deform a Lagrangian submanifold, arXiv preprint arXiv:dg- ga (1996)

    1996

  67. [75]

    Smoczyk,Mean curvature flow in higher codimension: introduction and survey, Global differential geometry, 2012, pp

    K. Smoczyk,Mean curvature flow in higher codimension: introduction and survey, Global differential geometry, 2012, pp. 231–274. MR3289845

    2012

  68. [76]

    Stavrou,Selfsimilar solutions to the mean curvature flow, J

    N. Stavrou,Selfsimilar solutions to the mean curvature flow, J. Reine Angew. Math.499 (1998), 189–198. MR1631112

    1998

  69. [77]

    Sun and Z

    A. Sun and Z. Wang,On mean curvature flow translators with prescribed ends, Arch. Ration. Mech. Anal.249(2025), no. 5, Paper No. 51, 60. MR4947201 CALIBRATION ENERGY AND MEAN CURVATURE FLOW 43

    2025

  70. [78]

    P. M. Topping,Nontrivial breathers for Ricci flow, Bull. Lond. Math. Soc.55(2023), no. 1, 344–348. MR4568346

    2023

  71. [79]

    Tsai, M.-P

    C.-J. Tsai, M.-P. Tsui, and M.-T. Wang,An ansatz for constructing explicit solutions of Hessian equations, arXiv preprint arXiv:2506.17701 (2025)

    2025

  72. [80]

    Tsai, M.-P

    C.-J. Tsai, M.-P. Tsui, and M.-T. Wang,Entire solutions of two-convex Lagrangian mean curvature flows, Calc. Var. Partial Differential Equations64(2025), no. 7, Paper No. 198,

    2025

  73. [81]

    H. Wang, H. Xu, and E. Zhao,A global pinching theorem for complete translating solitons of mean curvature flow, Pure Appl. Math. Q.12(2016), no. 4, 603–619. MR3842856

    2016

  74. [82]

    Wang,Uniqueness of self-similar shrinkers with asymptotically conical ends, J

    L. Wang,Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc.27(2014), no. 3, 613–638. MR3194490

    2014

  75. [83]

    Wang,Lectures on mean curvature flows in higher codimensions, Handbook of geo- metric analysis

    M.-T. Wang,Lectures on mean curvature flows in higher codimensions, Handbook of geo- metric analysis. No. 1, 2008, pp. 525–543. MR2483374

    2008

  76. [84]

    Warren,Nonpolynomial entire solutions toσ k equations, Comm

    M. Warren,Nonpolynomial entire solutions toσ k equations, Comm. Partial Differential Equa- tions41(2016), no. 5, 848–853. MR3508325

    2016

  77. [85]

    Y. L. Xin,Translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations54(2015), no. 2, 1995–2016. MR3396441

    2015

  78. [86]

    Yao,A uniqueness result for self-expanders with small entropy, Proc

    J. Yao,A uniqueness result for self-expanders with small entropy, Proc. Amer. Math. Soc. 150(2022), no. 6, 2695–2700. MR4399282

    2022

  79. [87]

    Yuan,A Bernstein problem for special Lagrangian equations, Invent

    Y. Yuan,A Bernstein problem for special Lagrangian equations, Invent. Math.150(2002), no. 1, 117–125. MR1930884

    2002

  80. [88]

    Yuan,Global solutions to special Lagrangian equations, Proc

    Y. Yuan,Global solutions to special Lagrangian equations, Proc. Amer. Math. Soc.134 (2006), no. 5, 1355–1358. MR2199179

    2006

  81. [89]

    Z¨ ust,A calibration inR 16 and Federer’s product question, arXiv preprint arXiv:2503.12402 (2025)

    R. Z¨ ust,A calibration inR 16 and Federer’s product question, arXiv preprint arXiv:2503.12402 (2025). (T. Miura)Department of Mathematics, Graduate School of Science, Kyoto Univer- sity, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan Email address:tatsuya.miura@mat...

    2025

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