The normal growth exponent of a codimension-1 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, codimension-1 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 bi-Lipschitz to hyperbolic $n$-space $\mathbb{H}^n$ and the normal growth exponent is at most 1, then $X$ is bi-Lipschitz 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, codimension-1 submanifold that is bi-Lipschitz 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.