Spectral Characterization of Poincaré–Einstein Manifolds with Infinity of Positive Yamabe Type

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 > 2. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold (X, g) is less than n/2 − 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(α) on the conformal infinity is nonnegative for all α ∈ [0,2].