A gluing approach for the fractional Yamabe problem with isolated singularities

Abstract

<jats:title>Abstract</jats:title><jats:p>We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an infinite-dimensional Lyapunov–Schmidt reduction method, that reduces the problem to an (infinite-dimensional) Toda-type system. The main technical part is the estimate of the interactions between different bubbles in the bubble towers.</jats:p>

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

“…The next development came in [4] for the fractional Yamabe problem with isolated singularities, that we have just mentioned. There the model for an isolated singularity is a Delaunay-type metric (see also [65,66,79] for the construction of constant mean curvature surfaces with Delaunay ends and [64,67] for the scalar curvature case).…”

ODE Methods in Non-Local Equations

“…There is an extensive literature on the fractional Yamabe problem by now. See [35,36,40,44] for the smooth case, [3,7,22,23] in the presence of isolated singularities, and [4,5,34] when the singularities are not isolated but a higher dimensional set.…”

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

“…When σ ∈ (0, 1) and n ≥ 2, the existence of Fowler solutions of (−∆) σ u = u n+2σ n−2σ was proved by DelaTorre-del Pino-González-Wei [15], which corresponds to σ ∈ (0, 1) and n ≥ 2 in (10). See also Ao-DelaTorre-González-Wei [1] for a gluing approach for the fractional Yamabe problem with multiple isolated singularities for σ ∈ (0, 1).…”