Minimal submanifolds and stability in Einstein manifolds

TL;DR

This paper analyzes the stability and spectral properties of minimal submanifolds in complex Einstein spaces, providing new computational methods and criteria for instability.

Contribution

It introduces a method to compute eigenvalues and stability criteria for minimal submanifolds in Einstein manifolds, including Berger spheres.

Findings

Computed index and nullity for minimal submanifolds in complex Einstein spaces

Provided an algorithm for higher eigenvalues of Laplacian on Berger spheres

Suggested criteria for instability of certain minimal submanifolds

Abstract

In this paper, we compute the index and nullity for minimal submanifolds of some complex Einstein spaces. We investigate the stability of these minimal submanifolds and suggest a criterion for instability for some cases. We also compute some higher eigenvalues for the Laplacian of the Berger spheres and provide with an algorithm.