CoChat
Sign Up
Cited by 6Geometric Analysis and Curvature FlowsAdvanced Differential Geometry ResearchAlgebraic Geometry and Number Theory

The DOI matches its Crossref record; the references it cites are not checked yet.

Ancient solutions and translators of Lagrangian mean curvature flow

Open access
Lotay, Jason D., Schulze, Felix, Székelyhidi, Gábor·2024·Publications Mathématiques de l'IHÉS
DOI 10.1007/s10240-023-00143-5
Open this paper in CoChat — free accountOpen PDF

Abstract

Suppose that ℳ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in 𝐂 n . We show that if ℳ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then ℳ is a translator. In particular in 𝐂 2 , all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.

Overview

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1007/s10240-023-00143-5

Cites

Statistics

Verification

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

The citation lists haven’t been fetched yet

References 25 · Cited by 6

Research works printed but not linked: 27

  1. [1.] Bogachev, V. I. Gaussian Measures, 62, Am. Math. Soc., Providence, 1998 · DOI
  2. [2.] Brendle, S. Ancient solutions to the Ricci flow in dimension 3, Acta Math., Volume 225 (2020), pp. 1-102 · DOI
  3. [3.] Brendle, S.; Choi, K. Uniqueness of convex ancient solutions to mean curvature flow in 𝐑 3 , Invent. Math., Volume 217 (2019), pp. 35-76 · DOI
  4. [4.] Choi, K.; Haslhofer, R.; Hershkovits, O. Ancient low entropy flows, mean convex neighborhoods, and uniqueness, Acta Math., Volume 228 (2022), pp. 217-301 · DOI
  5. [5.] Choi, K.; Haslhofer, R.; Hershkovits, O.; White, B. Ancient asymptotically cylindrical flows and applications, Invent. Math., Volume 229 (2022), pp. 139-241 · DOI
  6. [6.] Colding, T. H.; Minicozzi, W. P. II. Harmonic functions with polynomial growth, J. Differ. Geom., Volume 46 (1997), pp. 1-77 · DOI
  7. [7.] Colding, T. H.; Minicozzi, W. P. II. Generic mean curvature flow I: generic singularities, Ann. Math. (2), Volume 175 (2012), pp. 755-833 · DOI
  8. [8.] Colding, T. H.; Minicozzi, W. P. II. Complexity of parabolic systems, Publ. Math. Inst. Hautes Études Sci., Volume 132 (2020), pp. 83-135 · DOI
  9. [9.] Colding, T. H.; Minicozzi, W. P. Parabolic frequency on manifolds, Int. Math. Res. Not., Volume 2022 (2022), pp. 11878-11890 · DOI
  10. [10.] Daskalopoulos, P.; Hamilton, R.; Sesum, N. Classification of compact ancient solutions to the curve shortening flow, J. Differ. Geom., Volume 84 (2010), pp. 455-464 · DOI
  11. [11.] Ding, Y. An existence theorem of harmonic functions with polynomial growth, Proc. Am. Math. Soc., Volume 132 (2004), pp. 543-551 · DOI
  12. [12.] Donaldson, S.; Sun, S. Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II, J. Differ. Geom., Volume 107 (2017), pp. 327-371 · DOI
  13. [13.] Ecker, K. Logarithmic Sobolev inequalities on submanifolds of Euclidean space, J. Reine Angew. Math., Volume 522 (2000), pp. 105-118
  14. [14.] Ecker, K. Regularity Theory for Mean Curvature Flow, 57, Birkhäuser Boston, Boston, 2004 · DOI
  15. [15.] Ecker, K.; Huisken, G. Mean curvature evolution of entire graphs, Ann. Math. (2), Volume 130 (1989), pp. 453-471 · DOI
  16. [16.] Fukaya, K. Galois symmetry on Floer cohomology, Turk. J. Math., Volume 27 (2003), pp. 11-32
  17. [17.] Hamilton, R. S. Harnack estimate for the mean curvature flow, J. Differ. Geom., Volume 41 (1995), pp. 215-226 · DOI
  18. [18.] Huisken, G. Asymptotic behavior for singularities of the mean curvature flow, J. Differ. Geom., Volume 31 (1990), pp. 285-299 · DOI
  19. [19.] Ilmanen, T.; Neves, A.; Schulze, F. On short time existence for the planar network flow, J. Differ. Geom., Volume 111 (2019), pp. 39-89 · DOI
  20. [20.] Joyce, D. Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, EMS Surv. Math. Sci., Volume 2 (2015), pp. 1-62 · DOI
  21. [21.] Joyce, D.; Lee, Y.-I.; Tsui, M.-P. Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differ. Geom., Volume 84 (2010), pp. 127-161 · DOI
  22. [22.] Lambert, B.; Lotay, J. D.; Schulze, F. Ancient solutions in Lagrangian mean curvature flow, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 22 (2021), pp. 1169-1205
  23. [23.] Neves, A. Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math., Volume 168 (2007), pp. 449-484 · DOI
  24. [24.] Neves, A. Recent Progress on Singularities of Lagrangian Mean Curvature Flow, Surveys in Geometric Analysis and Relativity, 20, Int. Press, Somerville, 2011, pp. 413-438
  25. [25.] Simon, L. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math. (2), Volume 118 (1983), pp. 525-571 · DOI
  26. [27.] Thomas, R. P. Moment maps, monodromy and mirror manifolds, Symplectic Geometry and Mirror Symmetry (2001), pp. 467-498 · DOI
  27. [28.] Thomas, R. P.; Yau, S.-T. Special Lagrangians, stable bundles and mean curvature flow, Commun. Anal. Geom., Volume 10 (2002), pp. 1075-1113 · DOI

Metadata

Authors
Lotay, J. D., Schulze, F., & Székelyhidi, G.
Affiliations
  • University of Oxford (GB)
  • University of Warwick (GB)
  • Northwestern University (US)
Corresponding author
Lotay, Jason D.
ORCID
Venue
Publications Mathématiques de l'IHÉS
Publisher
MathDoc/Centre Mersenne
Published
2024-12-07
Volume
140
Pages
1-35
ISSN
  • 0073-8301 (ISSN-L)
  • 1618-1913
DOI
10.1007/s10240-023-00143-5
OpenAlex
W4390877439
Cited by
6
References
25
Citations by year
2026: 4 · 2025: 1
FWCI
1.60
Type
journal article
Language
en
Open access
Licence
Locations
  • Publications mathématiques de l IHÉS (publishedVersion, cc-by)
  • Oxford University Research Archive (ORA) (University of Oxford) (acceptedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Warwick Research Archive Portal (University of Warwick) (submittedVersion)
  • French digital mathematics library (Numdam) (submittedVersion)
  • arXiv (Cornell University) (cc-by)
Funders
  • National Science Foundation (1906216)
  • Leverhulme Trust (RPG-2016-174)
Topics
  • Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
  • Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
  • Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
Keywords
Lagrangian mean curvature flow, ancient solutions, translators, Lagrangian subspaces, blow-down, entropy, special Lagrangian submanifolds

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review

How to cite

Cite CoChat for the checks on this page, and the paper for its own findings.

This page
CoChat. (2026). Citation check: Ancient solutions and translators of Lagrangian mean curvature flow. https://app.cochat.ai/papers/10.1007/s10240-023-00143-5
The paper
Lotay, J. D., Schulze, F., & Székelyhidi, G. (2024). Ancient solutions and translators of Lagrangian mean curvature flow. Publications Mathématiques de l'IHÉS. https://doi.org/10.1007/s10240-023-00143-5