From Scattering Operators to Yamabe Equations

Jie Qing (UCSC)

We will talk on our recent works on the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. This family of equations is considered as a generalization of Yamabe equations. This family of conformally invariant equations is introduced via scattering operators on a Poincar\'{e} metrics associated with a conformal manifold. Particularly in the case of compact Kleinian manifolds such family of equations can be transformed into a family of integral equations. Then a recent method of moving plane in dealing with integral equations is adopted to derive the compactness of positive solutions to this family of equations. Therefore by regarding this family of equations as a deformation from the Yamabe equation we use a degree theory to produce positive solutions to each equation in this family.