---
OA_place: repository
OA_type: green
_id: '10011'
abstract:
- lang: eng
text: We propose a new weak solution concept for (two-phase) mean curvature flow
which enjoys both (unconditional) existence and (weak-strong) uniqueness properties.
These solutions are evolving varifolds, just as in Brakke's formulation, but are
coupled to the phase volumes by a simple transport equation. First, we show that,
in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461,
(1993)], any limit point of solutions to the Allen-Cahn equation is a varifold
solution in our sense. Second, we prove that any calibrated flow in the sense
of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean
curvature flow-is unique in the class of our new varifold solutions. This is in
sharp contrast to the case of Brakke flows, which a priori may disappear at any
given time and are therefore fatally non-unique. Finally, we propose an extension
of the solution concept to the multi-phase case which is at least guaranteed to
satisfy a weak-strong uniqueness principle.
acknowledgement: This project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 948819), and from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.
The content of this paper was developed and parts of it were written during a visit
of the first author to the Hausdorff Center of Mathematics (HCM), University of
Bonn. The hospitality and the support of HCM are gratefully acknowledged.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
- first_name: Tim
full_name: Laux, Tim
last_name: Laux
citation:
ama: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow:
Convergence of the Allen-Cahn equation and weak-strong uniqueness. Journal
of Differential Geometry. 2025;130:209-268. doi:10.4310/jdg/1747065796'
apa: 'Hensel, S., & Laux, T. (2025). A new varifold solution concept for mean
curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness.
Journal of Differential Geometry. International Press. https://doi.org/10.4310/jdg/1747065796'
chicago: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for
Mean Curvature Flow: Convergence of the Allen-Cahn Equation and Weak-Strong Uniqueness.”
Journal of Differential Geometry. International Press, 2025. https://doi.org/10.4310/jdg/1747065796.'
ieee: 'S. Hensel and T. Laux, “A new varifold solution concept for mean curvature
flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness,” Journal
of Differential Geometry, vol. 130. International Press, pp. 209–268, 2025.'
ista: 'Hensel S, Laux T. 2025. A new varifold solution concept for mean curvature
flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness. Journal
of Differential Geometry. 130, 209–268.'
mla: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean
Curvature Flow: Convergence of the Allen-Cahn Equation and Weak-Strong Uniqueness.”
Journal of Differential Geometry, vol. 130, International Press, 2025,
pp. 209–68, doi:10.4310/jdg/1747065796.'
short: S. Hensel, T. Laux, Journal of Differential Geometry 130 (2025) 209–268.
corr_author: '1'
date_created: 2021-09-13T12:17:10Z
date_published: 2025-05-01T00:00:00Z
date_updated: 2025-05-28T09:27:05Z
day: '01'
department:
- _id: JuFi
doi: 10.4310/jdg/1747065796
ec_funded: 1
external_id:
arxiv:
- '2109.04233'
intvolume: ' 130'
keyword:
- Mean curvature flow
- gradient flows
- varifolds
- weak solutions
- weak-strong uniqueness
- calibrated geometry
- gradient-flow calibrations
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2109.04233
month: '05'
oa: 1
oa_version: Preprint
page: 209-268
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: Journal of Differential Geometry
publication_identifier:
eissn:
- 1945-743X
issn:
- 0022-040X
publication_status: published
publisher: International Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'A new varifold solution concept for mean curvature flow: Convergence of the
Allen-Cahn equation and weak-strong uniqueness'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 130
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '17292'
abstract:
- lang: eng
text: The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkähler
4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of
the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian
isotopy classes of Lagrangians, and the Thomas-Yau conjecture on longtime existence
of the Lagrangian mean curvature ow. We also make observations concerning closed
geodesics, curve shortening flow and minimal surfaces.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jason D.
full_name: Lotay, Jason D.
last_name: Lotay
- first_name: Goncalo
full_name: Oliveira, Goncalo
id: 58abbde8-f455-11eb-a497-98c8fd71b905
last_name: Oliveira
citation:
ama: Lotay JD, Oliveira G. Special Lagrangians, Lagrangian mean curvature flow and
the Gibbons-Hawking ansatz. Journal of Differential Geometry. 2024;126(3):1121-1184.
doi:10.4310/jdg/1717348872
apa: Lotay, J. D., & Oliveira, G. (2024). Special Lagrangians, Lagrangian mean
curvature flow and the Gibbons-Hawking ansatz. Journal of Differential Geometry.
International Press. https://doi.org/10.4310/jdg/1717348872
chicago: Lotay, Jason D., and Goncalo Oliveira. “Special Lagrangians, Lagrangian
Mean Curvature Flow and the Gibbons-Hawking Ansatz.” Journal of Differential
Geometry. International Press, 2024. https://doi.org/10.4310/jdg/1717348872.
ieee: J. D. Lotay and G. Oliveira, “Special Lagrangians, Lagrangian mean curvature
flow and the Gibbons-Hawking ansatz,” Journal of Differential Geometry,
vol. 126, no. 3. International Press, pp. 1121–1184, 2024.
ista: Lotay JD, Oliveira G. 2024. Special Lagrangians, Lagrangian mean curvature
flow and the Gibbons-Hawking ansatz. Journal of Differential Geometry. 126(3),
1121–1184.
mla: Lotay, Jason D., and Goncalo Oliveira. “Special Lagrangians, Lagrangian Mean
Curvature Flow and the Gibbons-Hawking Ansatz.” Journal of Differential Geometry,
vol. 126, no. 3, International Press, 2024, pp. 1121–84, doi:10.4310/jdg/1717348872.
short: J.D. Lotay, G. Oliveira, Journal of Differential Geometry 126 (2024) 1121–1184.
corr_author: '1'
date_created: 2024-07-22T07:45:31Z
date_published: 2024-03-01T00:00:00Z
date_updated: 2025-09-08T08:27:51Z
day: '01'
department:
- _id: TaHa
doi: 10.4310/jdg/1717348872
external_id:
arxiv:
- '2002.10391'
isi:
- '001271790200007'
intvolume: ' 126'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2002.10391
month: '03'
oa: 1
oa_version: Preprint
page: 1121-1184
publication: Journal of Differential Geometry
publication_identifier:
issn:
- 0022-040X
publication_status: published
publisher: International Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking
ansatz
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 126
year: '2024'
...