Stability of the Cheng-yau Gradient Estimate

which is sharp as indicated in the Euclidean case. However even if M contains a small compact region where the Ricci curvature is not nonnegative, estimate (1.1) becomes very much different from (1.2) when r is large, due to the presence of the √ k term. Whether estimate (1.2) is stable under perturbation has been an open question for some time, in light of the known stability results on weaker properties of harmonic functions, such as the Harnack inequality. The goal of the paper is to confirm that (1.2) is stable when the nonpositive part of the Ricci curvature is sufficiently small in an integral sense. Let us mention that some smallness for the nonpositive part of the Ricci curvature is necessary for gradient estimate (1.2) to hold. For instance if the non-positive part of the Ricci curvature is so large that M admits a bounded nonconstant harmonic function, then clearly (1.2) can not hold. Throughout the paper ∆ is the Laplace-Beltrami operator, d(x, y) is the distance between x and y; and d(x) is the distance between x and a fixed reference point. |B(x, r)| denotes the volume of the geodesic ball of radius r centered at x.