The stable Bernstein theorem in $R^4$

I will discuss our second proof of the stable Bernstein theorem for minimal hypersurfaces in $R^4$: a complete, two-sided, stable minimal hypersurface in $R^4$ is flat. The proof relies on an intriguing relation between the stability inequality and the geometry of 3-manifolds with uniformly positive scalar curvature. If time permits, I will also talk about an extension to anisotropic minimal hypersurfaces. This is based on joint work with Otis Chodosh.