Smoothing Riemannian metrics (Q791894)

The purpose of this paper is to construct a smoothing operator which transforms Riemannian metrics with bounded curvature on compact manifolds into Riemannian metrics with essentially the same curvature bounds and, in addition, bounds on the derivatives of the curvature tensor. The construction is based on an evolution equation for Riemannian metrics. The main tools are interior regularity estimates for parabolic equations. Applications to almost flat manifolds and compactness properties of sets of Riemannian manifolds are outlined.