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 χ(M)≥0. Moreover, if χ(M)≠0, there exists a sequence of times t _k →∞, a double sequence of points $\{p_{k,l}\}_{l=1}^{N}$ and domains $\{U_{k,l}\}_{l=1}^{N}$ with p _k,l∈U _k,l satisfying the following:

• (i) $\mathrm{dist}_{g(t_{k})}(p_{k,l_{1}},p_{k,l_{2}})\rightarrow\infty$ as k→∞, for any fixed l₁≠l₂;

• (ii) for each l, (U _k,l,g(t _k ),p _k,l) converges in the $C_{\mathrm{loc}}^{\infty}$ sense to a complete negative Einstein manifold (M _∞,l ,g _∞,l ,p _∞,l ) when k→∞;

• (iii) $\operatorname {Vol}_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\rightarrow0$ as k→∞.