Index of minimal hypersurfaces in real projective spaces

TL;DR

The paper establishes sharp index bounds for minimal hypersurfaces in real projective spaces by extending sphere results through antipodal symmetry and orientation covers. It proves that every unstable one-sided minimal hypersurface in $\\mathbb{R}P^{n+1}$ has Morse index at least $n+2$, with equality attained by cubic isoparametric examples, and it computes explicit indices for Lawson surfaces to show the existence of closed embedded two-sided minimal surfaces of every odd index in $\\mathbb{R}P^3$. The approach combines Morse theory, spectral analysis of Jacobi operators, and symmetry arguments, including detailed handling of even/odd decompositions under involutions and leveraging Solomon’s eigenvalue computations. Collectively, the results provide sharp, constructive index bounds in projective spaces and give explicit constructions of minimal surfaces with prescribed instability properties, advancing understanding of stability, topology, and symmetry in minimal geometry.

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.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 2.3: Second variation formula
• Definition 3.1: Test functions $l_w$, $f_w$, $r_{v,w}$
• Lemma 3.2
• proof
• Lemma 3.3