Stable submanifolds in the product of projective spaces II

TL;DR

This paper characterizes stable compact minimal immersions in products of projective spaces, showing non-existence in certain cases and describing the structure of all stable immersions in others, extending previous results to higher dimensions.

Contribution

It generalizes known stability results for minimal immersions to arbitrary dimensions in products of complex, quaternionic, and octonionic projective spaces.

Findings

No odd-dimensional stable compact minimal immersions in products of two complex projective spaces.

Stable immersions in quaternionic cases are products of quaternionic projective subspaces and stable minimal immersions.

In octonionic cases, stable immersions are products of octonionic projective subspaces and stable minimal immersions.

Abstract

We prove that there do not exist odd-dimensional stable compact minimal immersions in the product of two complex projective spaces. We also prove that the only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold in the Riemmanian product. These generalize similar results of the second-named author of immersions with low dimensions or codimensions to immersions with arbitrary dimensions. In addition, we prove that the only stable compact minimal immersions in the product of a octonionic projective plane with any other Riemannian manifold are the products of octonionic projective subspaces with compact stable minimal immersions of the second manifold in the Riemmanian product.