The curvature of gradient Ricci solitons

Abstract

Our goal in this paper is to obtain further information about the curvature of gradient shrinking Ricci solitons. This is important for a better understanding and ultimately for the classification of these manifolds. The classification of gradient shrinkers is known in dimensions 2 and 3, and assuming locally conformally flatness, in all dimensions n ≥ 4 (see [14, 13, 6, 15, 20, 12, 2]). Many of the techniques used in these works required some control of the Ricci curvature. For example, in [15] gradient shrinking Ricci solitons which are locally conformally flat were classified assuming an integral condition on the Ricci tensor. This condition and other integral estimates of the curvature were later proved in [12]. Without making the strong assumption of being conformally flat, it is natural to ask whether similar estimates are true for the Riemann curvature tensor. In this paper we are able to prove pointwise estimates on the Riemann curvature, assuming in addition that the Ricci curvature is bounded. We will show that any gradient shrinking Ricci soliton with bounded Ricci curvature has Riemann curvature tensor growing at most polynomially in the distance function. We note that by Shi’s local derivative estimates we can then obtain growth estimates on all derivatives of the curvature. This, in particular, proves weighted L estimates for the Riemann curvature tensor and its covariant derivatives. We point out that for self shrinkers of the mean curvature flow Colding and Minicozzi [8] were able to prove weighted L estimates for the second fundamental form, assuming the mean curvature is positive. These estimates were instrumental in the classification of stable shrinkers. Our estimates can be viewed as parallel to theirs, however the classification of gradient Ricci solitons is still a major open question in the field. A gradient shrinking Ricci soliton is a Riemannian manifold (M, g) for which there exists a potential function f such that