Mass non-concentration at the nodal set and a sharp Wasserstein uncertainty principle

We prove $L^p$-mass concentration properties of Laplace eigenfunctions away from their nodal sets, extending a recent result in \cite{GM3} to all dimensions, and giving a slight refinement of a result in \cite{JN}. As a consequence, we are able to derive a sharp Wasserstein uncertainty principle that holds uniformly in the high frequency regime, proving a conjecture in \cite{St1}.