Curvature pinching based on integral norms of the curvature

Abstract A compact Riemannian manifold ( M, g ) of dimension 3 or higher admits a metric of constant (positive or negative) sectional curvature if the following conditions hold: the diameter is bounded from above, the part of the Ricci curvature which lies below some fixed negative number is bounded in L P norm for p > n /2, and the metric is almost spherical or almost hyperbolic in the L P sense. The idea of the proof is to obtain stronger (i.e. L ∞ ) pinching by deforming the initial metric using the Ricci flow, thus reducing the problem to the theorems of Gromov in the case r g < 0 and of Grove, Karcher and Ruh in the case r g > 0. The reduced curvature tensor changes along the flow according to the heat equation, which implies a weak nonlinear parabolic inequality for its norm. The iteration method of De Giorgi, Nash and Moser is applied to obtain the estimate for the maximum norm of the reduced curvature tensor. The crucial step in the iteration consists of controlling the Sobolev constant of the appropriate imbedding (which also changes along the flow, but behaves well) by the isoperimetric constant, which, in turn, can be bounded in terms independent of the particular manifold.