available in the non-unimodular setting. However, Schonmann [@schonmann] was able to avoid the use of the mass transport principle and showed:

\[schonmannsats\] Suppose $G$ is a transitive graph and that $p_c(G)<p_1<p_2\le 1$. Then any infinite cluster at level $p_2$ is a.s. $p_1$-stable.

Theorem \[schonmannsats\] has the following immediate consequence. Let $$p_u(G):=\inf\{p\,:\,\P_p-\mbox{a.s. there is a unique infinite
  cluster}\}$$ be the uniqueness treshold for percolation.

\[schonmanncor\] Suppose $G$ is a transitive graph and that $p>p_u(G)$. Then $\P_p[\mbox{there is a unique
    infinite cluster}]=1$.

So Corollary \[schonmanncor\] settles what happens above $p_u$. But there is also the question what happens at $p_u$. It turns out that the answer depends on the graph. The following theorem of Peres [@peres] is of special interest to us:

\[peressats\] Let $G=(V_G,E_G)$ and $H=(V_H,E_H)$ be two infinite transitive graphs and suppose $G$ is nonamenable and unimodular. Then at $p_u(G\times H)$ there is a.s. not a unique unbounded component.

In contrast to this result, Benjamini and Schramm [@itai1] showed that on any planar, transitive unimodular graph with one end, there is a.s. a unique infinite cluster at $p_u$.

We will now discuss analogues of Theorems \[hpsats\], \[schonmannsats\] and \[peressats\] in a continuum percolation setting. A Riemannian manifold $M$ is said 