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

2024/04/09 by María Fernanda Espinal, María del Mar González, Espinal, María Fernanda +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2404.05965

openalex publication_date 2024/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the σ₂--Yamabe equation, n>4, for solutions with a prescribed singular set Λ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than (n-√(n)-2)/2. The σ₂--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 σ₂ equation, which imply that the linearized operator has good mapping properties in weighted spaces.

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