<p>We resolve the long-standing problem of elucidating the cutoff phenomenon for a vast class of Markov processes, namely Markov diffusions with non-negative Bakry-Émery curvature. More precisely, we prove that any sequence of non-negatively curved diffusions exhibits cutoff in total variation as soon as the product condition is satisfied. Our result holds in Euclidean spaces as well as on Riemannian manifolds, and for arbitrary non-random initial conditions. Reversibility is not required. This simplifies, unifies and generalizes a number of works that have established cutoff through a delicate and model-dependent analysis of mixing times. The proof is elementary: we exploit a new differential relation between varentropy and entropy to produce a universal bound on the width of the mixing window.</p>