Quantitative C¹ - estimates on manifolds

We prove a $\mathsf{C}^1$-elliptic estimate of the form $ \sup_{B(x,r/2)} |\mathrm{grad} (\psi) | \leq C \left\{ \sup_{B(x,r)} |\Delta \psi| + \sup_{B(x,r)} |\psi| \right\}, $ valid on any complete Riemannian manifold $M$ and for any smooth function $\psi$ which is defined in a nighbourhood of $B(x,r)$, with an explicit quantitative control on the constant $C=C(B(x,r))$ in terms of the curvature of the geodesic ball $B(x,r)\subset M$.