| We obtain a local smoothing result for Riemannian manifolds with bounded Ricci curvature in dimension four. More precisely, given a Riemannian metric with bounded Ricci curvature and small L2-norm of curvature on a geodesic ball, we can find a smooth metric which is C1,alpha-close to the original metric on a smaller ball but still of definite size. Our result does not require a volume lower bound on the geodesic ball under consideration and hence can be applied to the collapsed case. As a consequence, the smaller geodesic ball involved is a quotient of a Euclidean ball by Euclidean isometries in the C 1,alpha-topology. Also we can deduce a definite bound on the Lp-norm of curvature for all p < infinity, which can be considered as a generalization of epsilon-regularity theorem for Einstein manifolds (J. Amer. Math. Soc. Vol. 19, No. 2 (2006), 487-525). |
