  Given a sequence of complete Riemannian manifolds $(M_n)$ of the same
dimension, we construct a complete Riemannian manifold $M$ such that for all $p
\in (1,\infty)$ the $L^p$-norm of the Riesz transform on $M$ dominates the
$L^p$-norm of the Riesz transform on $M_n$ for all $n$. Thus we establish the
following dichotomy: given $p$ and $d$, either there is a uniform $L^p$ bound
on the Riesz transform over all complete $d$-dimensional Riemannian manifolds,
or there exists a complete Riemannian manifold with Riesz transform unbounded
on $L^p$.
