Mean curvature flow of symmetric double graphs only develops singularities on the hyperplane of symmetry

TL;DR

This paper proves that symmetric double graphs evolving under mean curvature flow develop singularities only along the symmetry hyperplane, enabling solutions to free boundary problems with singular boundaries.

Contribution

It introduces a weak solution framework that preserves symmetry and singularity localization, and develops a notion called 'vanity' akin to convexity for analysis.

Findings

Singularities occur only on the hyperplane of symmetry.

Weak solutions can be approximated by smooth solutions in higher dimensions.

The approach facilitates solving free boundary problems with singular boundaries.

Abstract

By a symmetric double graph we mean a hypersurface which is mirror-symmetric and the two symmetric parts are graphs over the hyperplane of symmetry. We prove that there is a weak solution of mean curvature flow that preserves these properties and singularities only occur on the hyperplane of symmetry. The result can be used to construct smooth solutions to the free Neumann boundary problem on a supporting hyperplane with singular boundary. For the construction we introduce and investigate a notion named "vanity" and which is similar to convexity. Moreover, we rely on S\'aez' and Schn\"urer's "mean curvature flow without singularities" to approximate weak solutions with smooth graphical solutions in one dimension higher.