A note on the Compactness of Poincare-Einstein manifolds

TL;DR

This paper establishes a precise equivalence between the compactness of two types of compactifications of conformally compact Poincaré-Einstein manifolds with positive Yamabe conformal infinity, in specific function space topologies.

Contribution

It proves that the compactness of the standard and adapted compactifications are equivalent under certain boundary conditions and regularity assumptions.

Findings

Compactness of standard compactifications implies compactness of adapted ones.

The equivalence holds in specific $C^{k,eta}$ and $C^{l,eta}$ topologies.

The result applies to manifolds with positive Yamabe type conformal infinity.

Abstract

For a conformally compact Poincar\'{e}-Einstein manifold $(X,g_+)$, we consider two types of compactifications for it. One is $\bar{g}=\rho^2g_+$, where $\rho$ is a fixed smooth defining function; the other is the adapted (including Fefferman-Graham) compactification $\bar{g}_s=\rho^2_sg_+$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincar\'{e}-Einstein manifolds $\{(X, g_{+}^{(i)})\}$ with conformal infinity of positive Yamabe type, $\{\bar{g}^{(i)}\}$ is compact in $C^{k,\alpha}(\overline{X})$ topology if and only if $\{\bar{g}_s^{(i)}\}$ is compact in some $C^{l,\beta}(\overline{X})$ topology, provided that $\bar{g}^{(i)}|_{TM}=\bar{g}_s^{(i)}|_{TM}=\hat{g}^{(i)}$ and $\hat{g}^{(i)}$ has positive scalar curvature for each $i$. See Theorem 1.1 and Corollary 1.1 for the exact relation of $(k,\alpha)$ and $(l,\beta)$.