L\({}^ 2\)-curvature pinching (Q917054)

Let \((M^ n,g)\) be a compact Riemannian manifold and denote by d, r its diameter and the average scalar curvature, respectively. Suppose that for \(\Lambda >0\) we have \(d^ 2\cdot \max | K| \leq \Lambda^ 2,\) where K is the sectional curvature of M. In this paper the following three pinching theorems are proved: If there exists an \(\epsilon (n,\Lambda)>0\) such that a) \(r\neq 0\) and \((1/r^ 2)\int | Rm|^ 2<\epsilon (n,\Lambda)\) then M admits a metric of constant sectional curvature; \(b)\quad d^ 4\int | Rm|^ 2<\epsilon (n,\Lambda)\) then M is diffeomorphic to a compact quotient of a nilpotent Lie group by a discrete group of isometries; c) \(r<0\) and \(r^{-2}\int | Rc|^ 2<\epsilon (n,\Lambda)\) then M admits an Einstein metric of constant negative Ricci curvature. The basic method used to prove these theorems is to deform the metric in the direction of its Ricci curvature. The main idea is to show that a weak \(L^ 2\)-pinching assumption on some appropriate component of the curvature would lead after a short time along the flow of metrics, to a \(C^ 0\)-pinching condition for the same curvature component, thus reducing these results to known theorems. The technique used is the classical Moser iteration method together with some estimates for Sobolev constants and isoperimetric inequalities as obtained by \textit{S. Gallot} [C. R. Acad. Sci., Paris, Ser. I 296, 333-336 (1983; Zbl 0535.53034), and 365-368 (1983; Zbl 0535.53025)].