On the complexity of isometric immersions of hyperbolic spaces in any codimension

Abstract: Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a Lipschitz map $F:M^m\to\mathbb R^n$, where $M^m$ is a Hadamard manifold whose curvature lies between negative constants. The main result of this paper is that $F$ must perform a substantial compression: For every $r>0$ and integer $k\geq 2$ there exist $k$ geodesic balls of radius $r$ in $M^m$ that are arbitrarily far from each other, but whose images under $F$ are bunched together arbitrarily close in the Hausdorff sense of $\mathbb R^n$. In particular, every isometric embedding $\mathbb H^m\to\mathbb R^n$ of hyperbolic space must have a complex asymptotic behavior, regardless of how high the codimension is. Hence, there is no truly simple way to realize $\mathbb H^m$ isometrically inside any Euclidean space.