Stable submanifolds in the product of projective spaces

TL;DR

This paper classifies stable minimal immersions in products of projective spaces, identifying unique characteristics and providing a comprehensive understanding of their geometric properties.

Contribution

It offers a classification theorem for compact stable minimal immersions in products of projective spaces, highlighting the uniqueness of complex minimal immersions.

Findings

Characterization of complex minimal immersions as the only CSMI in products of two complex projective spaces.

Classification of CSMI in products involving complex or quaternionic projective spaces and compact rank one symmetric spaces.

Identification of conditions under which minimal immersions are stable in these geometric settings.

Abstract

We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension $1$ or dimension $1$ (codimension $1$ and $2$ or dimension $1$ and $2$) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension $2$ or dimension $2$ as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension $1$ or dimension $1$ (codimension $1$ and $2$ or dimension $1$ and $2$) in the product of a complex (quaternionic) projective space with any compact rank one symmetric space.