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arXiv · 2404.05965

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

María Fernanda Espinal·María del Mar González
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Abstract

In this paper we study the $\sigma_2$--Yamabe equation, $n>4$, for solutions with a prescribed singular set $\Lambda$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $(n-\sqrt{n}-2)/2$. The $\sigma_2$--curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method, used by Mazzeo-Pacard (JDG 1996) for the scalar curvature problem, can be used in the fully non-linear setting. This is a consequence of the conformal properties of the $\sigma_2$ equation, which imply that the linearized operator has good mapping properties in weighted spaces.

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María Fernanda Espinal, María del Mar González. 2024-04-09. A gluing construction of singular solutions for a fully non-linear equation in conformal geometry. https://arxiv.org/abs/2404.05965

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