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 $-b^2\le \kappa \le -1$. Suppose $\Sigma \subseteq X$ is a totally geodesic, codimension-1 submanifold and consider the geodesic flow ${\Phi }_t^{\nu }$ 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 \[ \lim_{t \rightarrow \pm \infty} \frac{ \Vert d Φ_t^ν\Vert_{\infty} }{ e^{β\vert t \vert}} < \infty, \] 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.

19 pages, 1 figure. Comments welcome!

References in corpus (3)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Systoles of hyperbolic 4-manifolds