Curvature effect in the spinorial Yamabe problem on product manifolds

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