Curvature effect in the spinorial Yamabe problem on product manifolds

Abstract

Let $(M_1,{\text{g}}^{(1)})$, $(M_2,{\text{g}}^{(2)})$ 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 ${\epsilon }^{-2}{\text{g}}^{(1)}\oplus {\text{g}}^{(2)}$, $\epsilon >0$. 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 $(M_1,{\text{g}}^{(1)})$.

38 Pages. arXiv admin note: text overlap with arXiv:2005.01448

References in corpus (1)

• Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity