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

2008/11/25 by Fang, Fuquan, Zhang, Yuguang, Zhang, Zhenlei
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0811.4028

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, χ(M)≠ 0, there exist a sequence 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 followings: [(i)] \dist_g(tk)(p_k,l1,p_k,l2)→∞ as k→∞, for any fixed l₁≠ l₂; [(ii)] for each l, (U_k,l,g(t_k),p_k,l) converges in the C_loc^∞ sense to a complete negative Einstein manifold (M_∞,l,g_∞,l,p_∞,l) when k→∞; [(iii)] \Vol_g(t_k)(M\backslash\bigcup_l=1^NU_k,l)→0 as k→∞.

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