Stable submanifolds in the product of projective spaces

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.

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