Fractional conformal Laplacians and fractional Yamabe problems

Abstract

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems observed by Chang and González, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf-type maximum principle together with interplay between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extends the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem.