Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume

Abstract

The main result of this paper shows that, if $g(t)$ is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold $M$ of finite volume, then the Euler characteristic number $\chi (M)\ge 0$. Moreover, $\chi (M)\ne 0$, there exist a sequence times $t_k\to \infty$, a double sequence of points $\{p_{k,l}{\}}_{l=1}^N$ and domains $\{U_{k,l}{\}}_{l=1}^N$ with $p_{k,l}\in U_{k,l}$ satisfying the followings: [(i)] $\dist_{g(t_k)}(p_{k,l_1},p_{k,l_2})\to\infty$ as $k\to \infty$, for any fixed $l_1\ne l_2$; [(ii)] for each $l$, $(U_{k,l},g(t_k),p_{k,l})$ converges in the $C_{loc}^{\infty }$ sense to a complete negative Einstein manifold $(M_{\infty ,l},g_{\infty ,l},p_{\infty ,l})$ when $k\to \infty$; [(iii)] $\Vol_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\to0$ as $k\to \infty$.

References in corpus (1)

• The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures