Spectral characterization of Poincaré-Einstein manifolds with infinity of positive Yamabe type

Abstract

In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only if the conformal infinity of $(X,g)$ is of positive Yamabe type. If this positivity is satisfied, we also show that the Green function of the fractional conformal Laplacian $P(\alpha )$ on the conformal infinity is non-negative for all $\alpha \in [0,2]$.

16 pages

References in corpus (2)

• Wave decay on convex co-compact hyperbolic manifolds
• Constant curvature foliations on asymptotically hyperbolic spaces