Isometric Embeddings of Conformally Compact Manifolds into Hyperbolic Spaces

TL;DR

This paper extends the Nash Embedding Theorem to conformally compact manifolds, showing they can be isometrically embedded into hyperbolic spaces of appropriate curvature, broadening understanding of geometric embeddings.

Contribution

It proves that conformally compact manifolds with negative curvature bounds can be embedded into hyperbolic spaces, analogous to Nash's theorem for Euclidean spaces.

Findings

Conformally compact manifolds can be embedded into hyperbolic spaces of matching curvature.

The embedding is transverse to the sphere at infinity.

The result generalizes Nash's theorem to a non-compact, conformally compact setting.

Abstract

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 $-\lambda^{2}$.