Min-max minimal hypersurfaces in higher dimensions

Abstract

In the recent decade, the Almgren-Pitts min-max theory has advanced the existence theory of minimal hypersurfaces in a closed Riemannian manifold $(M^{n+1}, g)$. When $3 \leq n+1 \leq 7$, many properties of these minimal hypersurfaces, such as areas, Morse indices, multiplicities, and spatial distributions, have been well studied. However, in higher dimensions ($n+1 \geq 8$), min-max minimal hypersurfaces may contain singularities. This phenomenon invalidates many helpful techniques in the low dimensions to investigate these geometric objects. I will show how one can utilize various deformation arguments to overcome the obstacles and prove generic abundance, index estimates, and most of the geometric properties of min-max minimal hypersurfaces. In particular, in dimension eight, en route to obtaining generic results, in joint work with Zhihan Wang, we prove generic regularity of minimal hypersurfaces.