Index of minimal hypersurfaces in real projective spaces

TL;DR

This paper establishes lower bounds on the Morse index of certain minimal hypersurfaces in real projective spaces and constructs examples with specific indices, advancing understanding of their stability properties.

Contribution

It proves a lower bound for the Morse index of unstable one-sided minimal hypersurfaces and constructs two-sided minimal surfaces with prescribed odd indices in real projective spaces.

Findings

Morse index of unstable one-sided minimal hypersurfaces is at least n+2

Cubic isoparametric minimal hypersurfaces attain this bound

Existence of two-sided minimal surfaces with each odd index in 3D real projective space

Abstract

We prove that for an embedded unstable one-sided minimal hypersurface of the $(n+1)$-dimensional real projective space, the Morse index is at least $n+2$, and this bound is attained by the cubic isoparametric minimal hypersurfaces. We also show that there exist closed embedded two-sided minimal surfaces in the 3-dimensional real projective space of each odd index by computing the index of the Lawson surfaces.