Let $V$ be a normal variety, and $p \in V$ a point. It is a theorem of Zariski[1] that the completion $\hat{\mathcal O}_{V,p}$ is a normal ring.
Does the converse also hold? Does $\hat{\mathcal O}_{V,p}$ normal, imply that $\mathcal O_{V,p}$ is normal as well?
[1] Zariski, Sur la normalité analytique des variétés normales, 1950
I’m turning my comment into an answer. $\hat{\mathcal{O}}_{V,p}$ is the completion of $\mathcal{O}_{V,p}$ (a Noetherian local ring) at its maximal ideal, and therefore is faithfully flat. By Stacks, Lemma 033G, if $\hat{\mathcal{O}}_{V,p}$ is normal, then so is $\mathcal{O}_{V,p}$.