Classification of singular solutions to the fractional Yamabe equation

Resumo: We study positive solutions of the critical, conformally invariant fractional Yamabe equation on the twice-punctured sphere and its Euclidean and cylindrical formulations. For fractional orders \(s\in (0,1)\) sufficiently close to one, we prove that every singular solution is a fractional Delaunay solution. The proof has two components: a local compactness argument applied to the normalized difference of two solutions, which yields convergence to a linear homogeneous limit; and a global classification step that uses sharp kernel bounds, weighted norms adapted to the Emden--Fowler transform, and an ODE-type barrier preventing nontrivial limits under noncompact recentering. A crucial step is obtaining a sharp supremum estimate from the Morse index, a property of the spherical bubble. We also establish the nondegeneracy of fractional Delaunay solutions for \(s \sim 1\) near one.

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