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

TL;DR

This paper proves that certain 3-spheres in complex 3-space are stable submanifolds with parallel mean curvature using spectral theory, highlighting their geometric stability properties.

Contribution

It demonstrates the stability of specific 3-spheres in complex space via spectral analysis, a novel approach in this geometric context.

Findings

The 3-sphere in a 2-dimensional complex subspace is $\Omega$-stable.

The submanifold has parallel mean curvature.

Spectral theory effectively proves stability in this setting.

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 a $\Omega$-stable submanifold with parallel mean curvature, when $\Omega$ is the K\"ahler calibration of rank 4 of $\mathbb{C}^3$.