Distance between minimal surfaces and flows
Differential Geometry
2026-05-12 v1 Analysis of PDEs
Abstract
We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.
Cite
@article{arxiv.2605.10589,
title = {Distance between minimal surfaces and flows},
author = {Tobias Holck Colding and William P. Minicozzi},
journal= {arXiv preprint arXiv:2605.10589},
year = {2026}
}