Poincare-Einstein 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.

17 pages

References in corpus (4)

• Estimates and Existence Results for some Fully Nonlinear Elliptic Equations on Riemannian Manifolds
• A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues
• Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds
• Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds