Fractional conformal Laplacians and fractional Yamabe problems

TL;DR

This paper explores fractional Yamabe problems using scattering operators and boundary value problems, revealing new maximum principles and conformal analysis phenomena that extend classical Yamabe problem insights.

Contribution

It introduces a novel formulation of fractional Yamabe problems linking scattering operators with boundary value problems, expanding the understanding of conformal geometry in fractional settings.

Findings

Established a Hopf type maximum principle for fractional Yamabe problems

Connected analysis of weighted trace Sobolev inequalities with conformal structures

Extended classical Yamabe phenomena to fractional and boundary contexts

Abstract

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