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Given a sequence of complete Riemannian manifolds $(M_n)$ of the samedimension, 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 thefollowing dichotomy: given $p$ and $d$, either there is a uniform $L^p$ boundon the Riesz transform over all complete $d$-dimensional Riemannian manifolds,or there exists a complete Riemannian manifold with Riesz transform unboundedon $L^p$.

A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds