Curvature effect in the spinorial Yamabe problem on product manifolds

Let $(M_1,{\mathit{\text{g}}}^{(1)})$, $(M_2,{\mathit{\text{g}}}^{(2)})$ be closed Riemannian spin manifolds. We study the existence of solutions of the spinorial Yamabe problem on the product $M_1\times M_2$ equipped with a family of metrics ${\epsilon }^{-2}{\mathit{\text{g}}}^{(1)}\oplus {\mathit{\text{g}}}^{(2)}$, $\epsilon >0$. Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor $M_1$, and which exhibit a spike layer as $\epsilon \to 0$. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $(M_1,{\mathit{\text{g}}}^{(1)})$.