Speaker: Felix Schulze (University of Warwick)

Title: Generic regularity for minimizing hypersurfaces in dimensions 9 and 10

Abstract:
Let $\Gamma$ be a smooth, closed (compact and boundaryless), oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$. The Plateau problem asks if among all smooth, compact, oriented hypersurfaces $M \subset \mathbb{R}^{n+1}$ with $\partial M = \Gamma$, does there exist one with least area? 

Foundational results in geometric measure theory can be used to show that for $n+1 \leq 7$ there is a smooth, compact, oriented, area-minimizing hypersurface $M$ solving the problem. In higher dimensions smooth minimizers can fail to exist but it is nevertheless known that away from a closed set $\text{sing} M \subset \mathbb{R}^{n+1}\setminus\Gamma$ of Hausdorff dimension $\leq n-7$, there is a minimizer $M$ which is a smooth hypersurface with boundary $\Gamma$.

A fundamental result of Hardt-Simon shows that the singularities (necessarily isolated points) for $n+1=8$ can be eliminated by a slight perturbation of the boundary, $\Gamma$, thus yielding solutions to the original problem. It has been a longstanding conjecture that similar results hold in higher dimensions. 

We show that for $n+1= 9,10$ singularities can still be perturbed away as well. This is joint work with O. Chodosh and C. Mantoulidis.

Zoom link: https://huji.zoom.us/j/88091075385?pwd=Q2IxRDBiYVY5Z2dFSEMvNjRMcWdYZz09