Generic Regularity of Minimal Hypersurfaces in Dimension 8

Abstract

In this paper, we show that every $8$-dimensional closed Riemmanian manifold with $C^{\infty }$-generic metrics admits a smooth minimal hypersurface. This generalized previous results by N. Smale and Chodosh-Liokumovich-Spolaor. Different from their local perturbation techniques, our construction is based on a global perturbation argument in and a novel geometric invariant which counts singular points with suitable weights.

27 pages: Some arguments simplified; References added; Comments welcome