Curvature effect in the spinorial Yamabe problem on product manifolds

TL;DR

This paper investigates the spinorial Yamabe problem on product manifolds, demonstrating the existence of solutions with spike layers influenced by the curvature of one factor as a parameter tends to zero.

Contribution

It introduces new existence results for solutions depending only on one factor and characterizes their asymptotic peak positions based on curvature.

Findings

Solutions depend only on $M_1$

Solutions exhibit spike layers as $ extit{ε} o 0$

Peak positions relate to curvature tensor

Abstract

Let $(M_1,\textit{g}^{(1)})$, $(M_2,\textit{g}^{(2)})$ 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 $\varepsilon^{-2}\textit{g}^{(1)}\oplus\textit{g}^{(2)}$, $\varepsilon>0$. 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 $\varepsilon\to0$. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $(M_1,\textit{g}^{(1)})$.