Stable $3$-Spheres in $\mathbb{C}^{3}$

Abstract

By only using spectral theory of the Laplace operator on spheres, we prove that the unit 3-dimensional sphere of a 2-dimensional complex subspace of $\mathbb{C}^3$ is an $\Omega$-stable submanifold with parallel mean curvature, when $\Omega$ is the K\"{a}hler calibration of rank $4$ of $\mathbb{C}^3$.