We construct some ODE-type solutions for the singular fractional Yamabe problem in conformal geometry. The fractional curvature, a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator constructed on the conformal infinity of a conformally compact Einstein manifold.

These ODE-type solutions are a generalization of the usual Delaunay and, in particular, solve the fractional Yamabe problem

$$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in} \ \r^n \backslash \{0\},$$

with an isolated singularity at the origin.

This is a fractional order ODE for which new tools need to be developed. The key of the proof is the computation of the fractional Laplacian in polar coordinates.

This is a joint work with Mar{\’}ia del Mar Gonz{\’}alez, Manuel del Pino and Juncheng Wei.

What happens if we remove a finite number of points? We will give the basic ideas for the construction of solutions in this case.

This is a joint work with Weiwei Ao, Mar{\’}ia del Mar Gonz{\’}alez and Juncheng Wei.