Maximum solutions of normalized Ricci flow on 4-manifolds

Abstract

We consider the maximum solution g(t), t is an element of [0, +infinity), to the normalized Ricci flow. Among other things, we prove that, if (M, omega) is a smooth compact symplectic 4-manifold such that b(2)(+) (M) > 1 and let g(t), t is an element of [0, infinity), be a solution to (1.3) on M whose Ricci curvature satisfies that vertical bar Ric(g(t))vertical bar <= 3 and additionally chi(M) = 3 tau(M) > 0, then there exists an m is an element of N, and a sequence of points {x(j,k) is an element of M}, j = 1, ... , m, satisfying that, by passing to a subsequence, (M, g(t(k) + t), x(1,k), ... . x(m,k)) -> (dGH) (coproduct(j=1) (m) N-j,N-g infinity, x(1,infinity), ... , x(m,infinity)), t is an element of [0,8), in the m-pointed Gromov-Hausdorff sense for any sequence t(k) -> infinity, where (N-j, g(infinity)), j = 1, ... , m, are complete complex hyperbolic orbifolds of complex dimension 2 with at most finitely many isolated orbifold points. Moreover, the convergence is C-infinity in the non-singular part of coproduct(m)(1) N-j and Vol(g0) (M) = Sigma(m)(j=1) Vol(g infinity)(N-j), where chi(M) (resp. tau(M)) is the Euler characteristic (resp. signature) of M.