transactions of the american mathematical society Volume 338, Number 2, August 1993 RICCI FLOW, EINSTEIN METRICS AND SPACE FORMS RUGANG YE Abstract. The main results in this paper are: ( 1 ) Ricci pinched stable Rie- mannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3) L2-pinched Riemannian manifolds can be deformed to space forms through Ricci flow. 0. Introduction and main results Einstein metrics on a compact manifold M of dimension « > 3 are char- acterized as critical points of the normalized total scalar curvature functional S on the space JZ of all (Riemannian) metrics on M. A natural procedure to construct Einstein metrics is therefore to deform an initial metric along the gradient flow of S. The explicit formula for S is this: for g eJZ, ^=vW=^\M^vg, where dvg is the volume form of g, V(g) = JM dvg, and Rg denotes the scalar curvature function of g . Simple computations [Sc] show that the gradient of S at g is given by _V{g)(2-n)/n (^RCg_ ^Rgg + ^Rgg^ , where Rcg denotes the Ricci tensor of g and Rg = Vig)~x JMRgdvg. As- suming w.l.o.g. Vig) = 1 at time t = 0 we can then write the gradient flow in the following way: <0-" U'-T' + n-ZrSR^ Here Tg = Rcg-Rgg/n is the traceless Ricci tensor of g and SRg = Rg-Rg . Along this flow the functional S would be increased. However, one observes [Sc] that an Einstein metric always minimizes S in its conformai class. Conse- quently one has to reverse the sign of the second term on the right-hand side of (0.1), which is the conformai component of the gradient. We keep the sign of Received by the editors November 14, 1990 and, in revised form, May 20, 1991. 1991 MathematicsSubject Classification. Primary 53C20; Secondary53C21, 53C25, 58G11, 58G25. Key words and phrases. Ricci flow, Einstein metrics, stability, space forms. Partially supported by NSF grant no. DMS87-03537. © 1993 American Mathematical Society 0002-9947/93 $1.00+ $.25 per page 871 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use