Manifolds of positive Ricci curvature with almost maximal volume

PERELMAN 10.In this note we consider complete Riemannian manifolds with Ricci curvature bounded from below.The well-known theorems of Myers and Bishop imply that a manifold M n with Ric ~ n -1 satisfies diam(1l1 n ) ~ diam(Sn(I)), Vol(Mn) ~ Vol(Sn(I)).It follows from Ch that equality in either of these estimates can be achieved only if M n is isometric to Sn (1).The natural conjecture is that a manifold M n with almost maximal diameter or volume must be a topological equivalent to Sn.With respect to diameter this is true only if M n satisfies some additional assumptions; see An, 0,GP,E.With respect to volume however no extra restriction is necesary.Theorem 1.For any integer n ~ 2 there exists an > 0 with the following property.Let M n be a complete Riemannian manifold with Ric ~ n -1.Suppose that Vol(Mn) ~ (1 -an) Vol(Sn(I)).Then M n is homeomorphic to Sn.In fact, we prove only that 7rj(Mn) = 0 for all i < n and refer to the work of Hamilton H for n = 3 and to the solution of generalized Poincare conjecture (Smale S, Freedman F) for n::j:.3 .Vanishing of homotopy groups is a simple consequence of the Main Lemma below.Its further simple corollaries are a noncompact version of Theorem 1 and a corresponding finiteness theorem (cf.P, Corollary B).Let BH (R) denote a ball of radius R in the simply connected space form of constant curvature H. Theorem 2. Let M n be a complete Riemannian manifold with Ric ~ 0; P EM. Suppose that Vol(BpTheorem 3.For any n, H, 91 ,R the set La (n, H, 91, R) of all complete Riemannian manifolds M n with diam(Mn) n~ 91, Ric ~ (n -I)H, and Vol(Bp (R)) ~ (1 -an) Vol(B H (R)) for all p E M n , contains only finitely many homotopy types.2 0 • Henceforward we fix n ~ 2 and ignore the dependence on n in our notations.We denote by M an arbitrary compact n-dimensional Riemannian manifold with Ric ~ n -1 ; all parameters below are supposed to be independent of M.