A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds

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$L^p$-normoftheRiesztransformon$M$dominatesthe$L^p$-normoftheRiesztransformon$M_n$forall$n$. 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 $L^p$.