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

F. Fontenele (Departamento de Geometria, Universidade Federal Fluminense, Niterói, RJ, Brazil)

F. Xavier (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.; and Department of Mathematics, Texas Christian University, Fort Worth, Tx., U.S.A.)

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

Received 18 July 2017

Published 9 February 2018