Stable submanifolds in the product of projective spaces

If $\Sigma_i$ is a stable submanifold of $M_i$, for $i = 1,2$, then $\Sigma_1\times\Sigma_2$ is a stable submanifold in $M_1 \times M_2$. Are all the stable submanifolds in $M_1 \times M_2$ like that? We will show that is the case for specific dimensions and codimensions in the product of a complex or quaternionic projective space with any other Riemannian manifold. We will also talk about the behaviour of stable submanifolds under a complex structure of the product of two complex projective spaces. We will describe how our proofs were motivated by work that has be done by Simons, Lawson, Ohnita, Torralbo and Urbano. Part of this work is joint with Shuli Chen (Stanford)