Spectral characterization of Poincar\'e-Einstein manifolds with infinity of positive Yamabe type

TL;DR

This paper provides a precise spectral criterion for conformally compact Einstein manifolds with positive Yamabe type at infinity, linking scattering poles to geometric positivity and analyzing the Green function of fractional conformal Laplacians.

Contribution

It establishes a sharp spectral characterization connecting scattering poles and Yamabe positivity for conformally compact Einstein manifolds.

Findings

Largest scattering pole less than (n/2 - 1) iff positive Yamabe type

Green function of fractional conformal Laplacian is non-negative for all alpha in [0,2]

Characterizes geometric positivity via spectral data

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