Ricci flow of almost non-negatively curved three manifolds
Ricci flow of almost non-negatively curved three manifolds
复制标题
DOI：
10.1515/crelle.2009.038
复制
发表时间：
2006-12
期刊：
影响因子：
--
通讯作者：
Miles Simon
中科院分区：
文献类型：
--
作者：
Miles Simon
摘要本文研究了几乎非负弯曲（可能是奇异）三维度量空间的Ricci流演化问题。我们考虑的非负弯曲度量空间是光滑黎曼流形（Mi, ig）的极限，i∈_1，其Ricci曲率大于-1 /i，其直径小于d0（与i无关），其体积大于v0 >（与i无关）。我们证明对于这样的空间，Ricci流的解在短时间t∈(0,t)存在，且解对于t∈(0,t)是光滑的，并且对于t∈(0,t)（对于某常数c = c(v 0, d 0, n)）具有Ricci (g(t))≧0和Riem (g(t))≧c/t。这允许我们对极限流形的拓扑类型和微分结构进行分类（根据Hamilton定理[J]）。[j] .数学学报，24(3):153-179,1986。
Abstract In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (Mi, ig), i ∈ ℕ, whose Ricci curvature is bigger than –1/i, and whose diameter is less than d 0 (independent of i) and whose volume is bigger than v 0 > 0 (independent of i). We show for such spaces, that a solution to Ricci flow exists for a short time t ∈ (0, T), that the solution is smooth for t > 0, and has Ricci (g(t)) ≧ 0 and Riem (g(t)) ≧ c/t for t ∈ (0, T) (for some constant c = c(v 0, d 0, n)). This allows us to classify the topological type and the differential structure of the limit manifold (in view of the theorem of Hamilton [J. Diff. Geom. 24: 153–179, 1986] on closed three manifolds with non-negative Ricci curvature).