PoincareEinstein metrics and the Schouten tensor
Abstract
We examine here the space of conformally compact metrics $g$ on the interior of a compact manifold with boundary which have the property that the $k^{th}$ elementary symmetric function of the Schouten tensor $A_g$ is constant. When $k=1$ this is equivalent to the familiar Yamabe problem, and the corresponding metrics are complete with constant negative scalar curvature. We show for every $k$ that the deformation theory for this problem is unobstructed, so in particular the set of conformal classes containing a solution of any one of these equations is open in the space of all conformal classes. We then observe that the common intersection of these solution spaces coincides with the space of conformally compact Einstein metrics, and hence this space is a finite intersection of closed analytic submanifolds.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2001
 arXiv:
 arXiv:math/0105171
 Bibcode:
 2001math......5171M
 Keywords:

 Differential Geometry;
 Analysis of PDEs;
 53C21;
 53C25;
 58D27
 EPrint:
 17 pages