负曲流形余维1超曲面的正规增长指数
The normal growth exponent of a codimension-1 hypersurface of a negatively curved manifold
摘要：
设$X$是具有收缩负曲率的Hadamard流形$-b^2\leqκ\leq -1$.设$Σ\subseteq X$是一个余维为1的全测地子流形，考虑$X$上的测地流$Φ^ν_t$由$Σ$上的单位法向量场$ν$生成.我们说$X$中$Σ$的正态增长指数至多为$β$，如果\[ \lim_{t \rightarrow \pm \infty} \frac{ \Vert d Φ_t^ν\Vert_{\infty} }{ e^{β\vert t \vert}} < \infty, \]其中$\Vert d Φ_t^ν\Vert_{\infty} $是$Σ$中所有点上$d Φ_t^ν$的算子范数的上确界。证明了如果$Σ$是双Lipschitz到双曲$n$-空间$\mathbb{H}^n$，且正态增长指数至多为1，则$X$是双Lipschitz到$\mathbb{H}^{n+1}$.作为应用，我们证明了：如果$M$是一个闭的负弯曲的$(n+1)$-流形，$N\subset M$是一个全测地的余维1子流形，它是一个双Lipschitz双曲流形，并且它的法向增长指数至多为1，那么$π_1(M)$同构于$\text{Isom}(\mathbb{H}^{n+1})$中的一个格.最后，我们表明，正常的增长指数的假设是必要的，在尺寸至少为4。
英文摘要：
Let $X$ be a Hadamard manifold with pinched negative curvature $-b^2\leqκ\leq -1$. Suppose $Σ\subseteq X$ is a totally geodesic, codimension-1 submanifold and consider the geodesic flow $Φ^ν_t$ on $X$ generated by a unit normal vector field $ν$ on $Σ$. We say the normal growth exponent of $Σ$ in $X$ is at most $β$ if \[ \lim_{t \rightarrow \pm \infty} \frac{ \Vert d Φ_t^ν\Vert_{\infty} }{ e^{β\vert t \vert}} < \infty, \] where $\Vert d Φ_t^ν\Vert_{\infty} $ is the supremum of the operator norm of $d Φ_t^ν$ over all points of $Σ$. We show that if $Σ$ 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 $π_1(M)$ 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.
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