@article{10011,
  abstract     = {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.},
  author       = {Hensel, Sebastian and Laux, Tim},
  issn         = {1945-743X},
  journal      = {Journal of Differential Geometry},
  keywords     = {Mean curvature flow, gradient flows, varifolds, weak solutions, weak-strong uniqueness, calibrated geometry, gradient-flow calibrations},
  pages        = {209--268},
  publisher    = {International Press},
  title        = {{A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness}},
  doi          = {10.4310/jdg/1747065796},
  volume       = {130},
  year         = {2025},
}

@article{17292,
  abstract     = {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.},
  author       = {Lotay, Jason D. and Oliveira, Goncalo},
  issn         = {0022-040X},
  journal      = {Journal of Differential Geometry},
  number       = {3},
  pages        = {1121--1184},
  publisher    = {International Press},
  title        = {{Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz}},
  doi          = {10.4310/jdg/1717348872},
  volume       = {126},
  year         = {2024},
}