Some aspects of the variational nature of mean curvature flow

Abstract

We show that the classical solution of the heat equation can be seen as the minimizer of a suitable functional defined in space-time. Using similar ideas, we introduce a functional $\mathcal{F}$ on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with $\mathcal{F}$. We show some connections between minimizers of $\mathcal{F}$ and mean curvature flow.