Constant Q-curvature metrics with Delaunay ends: the nondegenerate case

TL;DR

This paper constructs a family of solutions to the positive singular Q-curvature problem on compact manifolds with punctures, using advanced perturbation and gluing techniques, especially in higher dimensions with specific geometric conditions.

Contribution

It introduces a novel method for solving the singular Q-curvature problem on nondegenerate manifolds with punctures, extending previous results to higher dimensions with Weyl tensor conditions.

Findings

Constructed solutions on manifolds with punctures

Developed techniques to control convergence of the Paneitz operator

Addressed matching of interior and exterior solutions without geometric Jacobi fields

Abstract

We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that the Weyl tensor vanishes to sufficiently high order at the singular points. On a technical level, we use perturbation methods and gluing techniques based on the mapping properties of the linearized operator both in a small ball around each singular point and in its exterior. Main difficulties in our construction include controlling the convergence rate of the Paneitz operator to the flat bi-Laplacian in conformal normal coordinates and matching the Cauchy data of the interior and exterior solutions; the latter difficulty arises from the lack of geometric Jacobi fields in the kernel of the linearized operator. We overcome both these difficulties by constructing suitable auxiliary functions.