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

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.