The fractional Yamabe problem with singularities: from zero to maximal dimension

Dr Azahara de la Torre Pedraza (Universidad de Granada) will present her research in Analysis.

The so-called Yamabe problem in Conformal Geometry is 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 nonzero-dimensional singularities can be transformed into reducing the non-linearity to a Sobolev-subcritical power. A quite recent notion of non-local curvature gives rise to a parallel study which weakens the geometric assumptions giving rise to a non-local semilinear elliptic (Sobolev-critical) PDE. 

In this talk, we will focus on the Euclidean space in the presence of singularities of maximal possible dimension. In both cases, local and non-local, we will construct singular solutions of a local/non-local semilinear elliptic equation with superlinear nonlinearity which is subritical for Sobolev embeddings, but it is critical for the existence of singular solutions (below such threshold the singularity is removable).

This is a joint work with H. Chan.

The talks are held using Zoom. Anyone interested in attending (outside the School of Mathematics) should email Ben Sharp at to request zoom coordinates