Stability of Homogeneous minimal hypersurfaces in the Page space and $Y^{p,q}$ Sasaki-Einstein manifolds

TL;DR

This paper studies the stability of homogeneous minimal hypersurfaces in specific Einstein manifolds, explicitly computing their spectra and indices to understand their geometric stability properties.

Contribution

It provides a complete classification of homogeneous minimal hypersurfaces in the Page space and $Y^{p,q}$ manifolds, including spectral analysis and stability indices.

Findings

All homogeneous minimal hypersurfaces are classified.

Explicit spectra of stability operators are computed.

Indices of the hypersurfaces are determined.

Abstract

We investigate the stability of homogeneous minimal submanifolds in two families of closed Einstein manifolds, the Page space $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ and the Sasaki-Einstein spaces $Y^{p,q}$, which are equipped with cohomogeneity-one Einstein metrics admitting the isometric action of $SU(2) \times U(1)$ and $U(1) \times U(1) \times SU(2)$ respectively. We determine all the homogeneous, minimal hypersurfaces and explicitly compute the spectrum of their associated stability operators and determine their index.