On the simply connectedness of non-negatively curved K\
On the simply connectedness of non-negatively curved K\
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DOI：
10.1090/s0002-9947-2011-05223-2
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发表时间：
2008-06
期刊：
影响因子：
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通讯作者：
Albert Chau;Luen-Fai Tam
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作者：
Albert Chau;Luen-Fai Tam
研究了具有一致有界非负全纯双截曲率的Kahler-Ricci流的完全非紧长时间解（M，g（t））.我们将证明当Ricci曲率为正且一致收缩时，即当$ R_\ijb \ge cRg_\ijb$ at $（p，t）$对所有$t$且$c>0$时，则在可能沿沿着某个序列$t_i \到\infty$重新标度$g（t）$后，在$p$附近总是存在局部梯度K\“ahler Ricci孤子极限.作为直接推论，我们将证明g（t）沿着$t_i$的内射半径从下沿着$t_i $一致有界，因此$M$实际上必须是单连通的。关于$M$和不动点的全纯等距群的一致化的其他结果也将建立。然后，我们将考虑移除对于$（M，g（t））$的正Ricci的条件。结合Cao对KahlerRicci流Cao 04的分裂和Ni-Tam 03的技巧，我们证明了当Ricci曲率的正特征值在M$中的某点$p一致收缩时，$M$具有特殊的全纯纤维丛结构.我们将讨论一种特殊情况，即具有非负全纯双截平均二次曲率衰减的完备K“ahler流形以及稳定梯度K“ahler Ricci孤子.
We study complete noncompact long time solutions $(M, g(t))$ to the K\"ahler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\ijb \ge cRg_\ijb$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient K\"ahler Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \to \infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ must in fact be simply connected. Additional results concerning the uniformization of $M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci for $(M, g(t))$. Combining our results with Cao's splitting for K\"ahler Ricci flow \cite{Cao04} and techniques of Ni-Tam \cite{NiTam03}, we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \in M$, then $M$ has a special holomorphic fiber bundle structure. We will treat a special cases, complete K\"ahler manifolds with non-negative holomorphic bisectional and average quadratic curvature decay as well as the case of steady gradient K\"ahler Ricci solitons.