发表机构
Hong Kong University of Science and Technology; Texas Tech University（香港科技大学; 德克萨斯理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究紧致流形上有限时间Kähler-Ricci流的奇点类型，完成双螺栓、螺母-螺栓、双螺母三类情形的分析，证明其奇点为I型并得到极限模型结构。
AI 中文摘要
我们研究紧致流形上Kähler-Ricci流的奇点类型与模型，这类紧致流形由Kähler-Einstein流形的乘积$N:= N_1 \times \boldsymbol{\times} N_r$上的圆丛的1参数叶状结构构造，其度量采用\ucf2011、WW等文献中考虑的假设形式。在作者早期的工作\ucf20中，我们研究了“双螺栓”情形，即叶状结构的两端均以“螺栓”$N$闭合。本文中，我们继续研究更微妙的“螺母-螺栓”和“双螺母”情形：前者的区间一端以螺母型坍缩（即$N':= N_2 \times \boldsymbol{\times} N_r$）闭合，另一端以螺栓（即$N$）闭合，紧致化空间$\boldsymbol{\times}M$是$N'$上的$\boldsymbol{CP}^{m+1}$丛；“双螺母”情形的两端均以螺母型坍缩闭合，此时必有两个$N_i$分别为$\boldsymbol{CP}^{m_0}$和$\boldsymbol{CP}^{m_\boldsymbol{\times}}$，紧致化空间$\boldsymbol{\times}M$是$\boldsymbol{CP}^{m_0+m_\boldsymbol{\times}+1}$在$\boldsymbol{k\boldsymbol{\times}3}N_k$上的丛。我们证明，在“双螺栓”“螺母-螺栓”和“双螺母”所有情形中，奇点必为I型；此外，我们研究了三种情形下流的重标度与膨胀序列的有标记Cheeger-Gromov极限，证明极限模型必为$(\boldsymbol{\times}^{m+1}, g_\boldsymbol{\times}(t)) \times (\boldsymbol{C}^{k}, \textrm{flat})$，其中$m,k \boldsymbol{\times}0$，$\boldsymbol{\times}$为以下之一：$\boldsymbol{CP}^{m+1}$、$\textrm{Tot}(\boldsymbol{L}^{\boldsymbol{\times}(m+1)})$，或满足$m_0 + m_\boldsymbol{\times} = m$的射影化$\boldsymbol{P}\big(\boldsymbol{O}^{\boldsymbol{\times}(m_0+1)} \boldsymbol{\times} \boldsymbol{L}^{\boldsymbol{\times}(m_\boldsymbol{\times}+1)}\big)$，$\boldsymbol{L}$是$N_1, \boldsymbol{\times}, N_r$中部分因子乘积上的线丛；度量$g_\boldsymbol{\times}(t)$是满足圆丛假设的Kähler-Ricci收缩子。
英文摘要
We study the singularity type and models of the Kähler--Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler--Einstein manifolds $N := N_1 \times \cdots \times N_r$, with metric constructed using the ansatz considered in \cite{DW2011}, \cite{WW} et. al. In the earlier work \cite{FT} by the authors, we considered the ``two-bolt'' case where both ends of the foliation close with the ``bolt'' $N$. In this article, we continue our work on the more subtle ``nut-bolt'' and ``two-nut'' cases. The former has one end of the interval closes with a nut-type collapse (i.e. $N' := N_2 \times \cdots \times N_r$) and the other with a bolt (i.e. $N$). The compactification $\widehat{M}$ is then a $\mathbb{CP}^{m+1}$-bundle over $N'$. The ``two-nut'' case is one that both ends close with nut-type collapses, necessarily two of the $N_i$'s must be $\mathbb{CP}^{m_0}$ and $\mathbb{CP}^{m_\ell}$, and the compactification $\widehat{M}$ is a $\mathbb{CP}^{m_0+m_\ell+1}$-bundle over $\prod_{k\geq 3}N_k$. We proved that in all ``two-bolt'', ``nut-bolt'' and ``two-nut'' caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be $(Σ^{m+1}, g_Σ(t)) \times (\mathbb{C}^{k}, \textrm{flat})$ with $m, k \geq 0$, where $Σ$ is one of the following: $\mathbb{CP}^{m+1}$, $\textrm{Tot}(\mathcal{L}^{\oplus(m+1)})$, or a projectivization $\mathbb{P}\big(\mathcal{O}^{\oplus(m_0+1)} \oplus \mathcal{L}^{\oplus(m_\ell+1)}\big)$ with $m_0 + m_\ell = m$, and $\mathcal{L}$ is a line bundle over the product of \emph{some} of the $N_1, \cdots, N_r$ factors. The metric $g_Σ(t)$ is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.
Comments47 pages; comments are welcome
