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arXiv · 2301.04934

Curvature effect in the spinorial Yamabe problem on product manifolds

Thomas Bartsch·Tian Xu
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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 $(M_1,\textit{g}^{(1)})$.

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Thomas Bartsch, Tian Xu. 2023-01-12. Curvature effect in the spinorial Yamabe problem on product manifolds. https://doi.org/10.1007/s00526-022-02303-7

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