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arXiv · 2605.10589

Distance between minimal surfaces and flows

Tobias Holck Colding·William P. Minicozzi II
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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.

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Tobias Holck Colding, William P. Minicozzi II. 2026-05-11. Distance between minimal surfaces and flows. https://arxiv.org/abs/2605.10589

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