Solutions to the fractional Yamabe problem with singularities: Construction and properties.

Author: Azahara de la Torre

Abstract: The so-called Yamabe problem in Conformal Geometry consists in finding a metric conformal to a given one and which has constant scalar curvature. From the analytic point of view, this problem becomes a semilinear elliptic PDE with critical (for the Sobolev embedding) power non-linearity. If we study the problem in the Euclidean space, allowing the presence of positive-dimensional singularities can be transformed into reducing the non-linearity to a Sobolev-subcritical power. When, instead of scalar, we work with non-local curvature we need to deal with the pertinent non-local semi-linear elliptic PDE.

In this talk, we will focus on the study of the solutions of these non-local (or fractional) semilinear elliptic PDEs which represent metrics which are singular. In collaboration with Ao, Chan, Fontelos, González and Wei, we covered the construction of solutions which are singular along (zero and positive-dimensional) smooth submanifolds. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of ``maximal'' dimension for the singularity was not covered. In collaboration with H. Chan, we covered the construction for this specific dimension using asymptotic analysis and the stability and semi-linearity of the equation. Moreover, we developed new methods to study the singular behaviour of the solutions. In a recent work, in collaboration with S. Cruz-Blázquez and D. Ruiz, we provide qualitative properties for all the solutions in the isolated singularity-case.