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

TL;DR

This paper proves that complete non-singular solutions of the normalized Ricci flow on noncompact finite volume 4-manifolds imply nonnegative Euler characteristic and describe their asymptotic geometric structure.

Contribution

It establishes a link between Ricci flow solutions and the topological and geometric structure of noncompact manifolds, including convergence to Einstein manifolds.

Findings

Euler characteristic of the manifold is nonnegative.

Existence of sequences where the manifold converges to Einstein manifolds.

Remaining volume tends to zero outside certain domains.

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 $\chi(M)\geq0$. Moreover, $\chi(M)\neq 0$, 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 $l_1\neq l_2$; [(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 $k\to\infty$.