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

TL;DR

This work analyzes the stability of homogeneous minimal hypersurfaces in two families of positive Einstein manifolds, the Page space and the Sasaki-Einstein spaces $Y^{p,q}$. By exploiting cohomogeneity-one symmetry, the authors reduce the Jacobi stability problem to explicit spectral calculations for the principal orbits, yielding exact stability spectra. For the Page space, there is a single homogeneous minimal hypersurface, which is totally geodesic and has index $1$; for $Y^{p,q}$, there is a single homogeneous minimal hypersurface with index $3$ and not totally geodesic. These results provide precise stability data for natural minimal hypersurfaces in Page-type and Sasaki-Einstein geometries, with potential implications for geometric analysis and AdS/CFT-related contexts.

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.

Figures (1)

• Figure 1: $\bar{y}$ as a function of $\epsilon = q/p \in (0,1)$. As $\epsilon \to 1$, $\bar{y} \to -1/8$.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 3.1
• proof