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# Pure and Applied Mathematics Quarterly

## Volume 12 (2016)

### Number 2

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

Pages: 243 – 259

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n2.a3

#### Authors

#### 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 $r \gt 0, \epsilon \gt 0$ and integer $k \geq 2$ there exist $k$ geodesic balls of radius $r$ in $M^m$ that are at least $\epsilon^{-1}$ apart, but whose images under $F$ are $\epsilon$-close in the Hausdorff sense of $\mathbb{R}^n$.* In particular, any isometric embedding $\mathbb{H}^m \to \mathbb{R}^n$ of hyperbolic space, proper or not, must have a rather complex asymptotic behavior, no matter how high the codimension $n - m$ is allowed to be.

#### Keywords

isometric embeddings of hyperbolic spaces, Lipschitz map, Hadamard manifold, Nash theorem

The first author’s work was partially supported by CNPq (Brazil).

Received 18 July 2017

Published 9 February 2018