The normal growth exponent of a codimension1 hypersurface of a negatively curved manifold
Abstract
Let $X$ be a Hadamard manifold with pinched negative curvature $b^2\leq\kappa\leq 1$. Suppose $\Sigma\subseteq X$ is a totally geodesic, codimension1 submanifold and consider the geodesic flow $\Phi^\nu_t$ on $X$ generated by a unit normal vector field $\nu$ on $\Sigma$. We say the normal growth exponent of $\Sigma$ in $X$ is at most $\beta$ if \[ \lim_{t \rightarrow \pm \infty} \frac{ \Vert d \Phi_t^\nu \Vert_{\infty} }{ e^{\beta \vert t \vert}} < \infty, \] where $\Vert d \Phi_t^\nu \Vert_{\infty} $ is the supremum of the operator norm of $d \Phi_t^\nu $ over all points of $\Sigma$. We show that if $\Sigma$ is biLipschitz to hyperbolic $n$space $\mathbb{H}^n$ and the normal growth exponent is at most 1, then $X$ is biLipschitz to $\mathbb{H}^{n+1}$. As an application, we prove that if $M$ is a closed, negatively curved $(n+1)$manifold, and $N\subset M$ is a totally geodesic, codimension1 submanifold that is biLipschitz to a hyperbolic manifold and whose normal growth exponent is at most 1, then $\pi_1(M)$ is isomorphic to a lattice in $\text{Isom}(\mathbb{H}^{n+1})$. Finally, we show that the assumption on the normal growth exponent is necessary in dimensions at least 4.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.06149
 Bibcode:
 2021arXiv210906149B
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Differential Geometry;
 Mathematics  Group Theory;
 20F65;
 37D40;
 53C20
 EPrint:
 19 pages, 1 figure. Comments welcome!