Maximum solutions of normalized Ricci flows on 4-manifolds

Abstract

We consider maximum solution $g(t)$, $t\in [0,+\infty )$, 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 $g(t),t\in [0,\infty )$, be a solution to (1.3) on $M$ whose Ricci curvature satisfies that $|\text{Ric}(g(t))|\le 3$ and additionally $\chi (M)=3\tau (M)>0$, then there exists an $m\in \mathbb{N}$, and a sequence of points $\{x_{j,k}\in M\}$, $j=1,...,m$, satisfying that, by passing to a subsequence, $$(M,g(t_k+t),x_{1,k},...,x_{m,k})\stackrel{d_{GH}}{⟶}(\coprod _{j=1}^m{N}_j,g_{\infty },x_{1,\infty },...,,x_{m,\infty }),$$ $t\in [0,\infty )$, in the $m$-pointed Gromov-Hausdorff sense for any sequence $t_k⟶\infty$, where $(N_j,g_{\infty })$, $j=1,...,m$, are complete complex hyperbolic orbifolds of complex dimension 2 with at most finitely many isolated orbifold points. Moreover, the convergence is $C^{\infty }$ in the non-singular part of ${\coprod }_1^m{N}_j$ and ${\text{Vol}}_{g_0}(M)={\sum }_{j=1}^m{\text{Vol}}_{g_{\infty }}(N_j)$, where $\chi (M)$ (resp. $\tau (M)$) is the Euler characteristic (resp. signature) of $M$.

23 pages