The celebrated Nash Embedding Theorem asserts that every closed Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. In this paper, we prove an analogous result in the conformally compact context. Let $\left(M,g\right)$ be a conformally compact manifold whose sectional curvature at infinity is strictly bounded below by a negative constant $-\lambda^{ 2}$. We prove that $\left(M,g\right)$ can be realized as a submanifold, transverse to the sphere at infinity, of a sufficiently high-dimensional rescaled hyperbolic space of constant curvature $-\lambda^{ 2}$.
Categories: math.DG