De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

Tim Laux; Andrea Poiatti

De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

Analysis of PDEs 2026-07-04 v1 Differential Geometry

Authors: Tim Laux , Andrea Poiatti

Abstract

We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate $L^2$ distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

Keywords

mean curvature flow varifold solutions minimizing movements De Giorgi weak solutions variational methods

Cite

@article{arxiv.2607.03930,
  title  = {De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach},
  author = {Tim Laux and Andrea Poiatti},
  journal= {arXiv preprint arXiv:2607.03930},
  year   = {2026}
}