Abstract

Critical points of the area functional are known as minimal surfaces, and in the case of graphs over domains in Euclidean space they satisfy the so-called minimal surface equation. In high dimensions it is possible for area minimising surfaces to be singular, as illustrated by Simons cone. In this talk we will discuss several results surrounding the singular set of minimal hypersurfaces, including Allard’s regularity theorem and dimension bounds. To establish bounds on the dimension of the singular set we will study the possible tangent cones to the hypersurface, which are formed by taking weak limits when “blowing up” the hypersurface at a given point. This is done by establishing monotonicity formulae, compactness results, and the non-existence of certain cones in low dimensions. Time permitting, we will mention how such techniques can be used to prove the Bernstein theorem, which tells us that entire minimal graphs in dimensions less than 8 are in fact planar.