Distance between minimal surfaces and flows

TL;DR

This paper demonstrates that the distance between minimal hypersurfaces functions as a Lipschitz continuous supersolution of an elliptic PDE, providing new insights and estimates, especially when hypersurfaces evolve by mean curvature flow.

Contribution

It introduces a novel PDE-based framework for analyzing the distance between minimal and evolving hypersurfaces, with implications for geometric analysis.

Findings

Distance is a Lipschitz continuous supersolution of an elliptic PDE.

Provides local Harnack inequalities for the distance under mean curvature flow.

Extends to a fully parabolic setting for two evolving hypersurfaces.

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.