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

Colin Guillarmou (1) , Jie Qing (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 Dept of Mathematics

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 $\alpha\in [0, 2]$.

Domaines: Géométrie différentielle [math.DG]

Colin Guillarmou, Jie Qing. Spectral characterization of Poincaré-Einstein manifolds with infinity of positive Yamabe type. International Mathematics Research Notices, 2010, 2010, pp.1720-1740. ⟨hal-00418128⟩