Extinction profile of complete non-compact solutions to the Yamabe flow
Extinction profile of complete non-compact solutions to the Yamabe flow
复制标题
Yamabe 流的完整非紧凑解的消光曲线
DOI：
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发表时间：
2013
影响因子：
0.7
通讯作者：
N. Šešum
中科院分区：
文献类型：
--
作者：
P. Daskalopoulos;John King;N. Šešum
本文研究了共形平坦Yamabe流完全非紧解的{em奇点形成}，该流的共形因子在无穷远处具有{em柱面行为}。它们的奇异性轮廓恰好是{em Yamabe孤子}，这是演化度规的共形因子所满足的快扩散方程的{em自相似解}。自相似轮廓由初始数据在无穷远处的二阶渐近性确定，该渐近性与相应的自相似解的渐近性相匹配。解可能在圆柱形尾的灭绝时间T$处灭绝，也可能存在时间超过T$。在第一种情况下，奇异性轮廓由{em Yamabe shrinker}描述，该收缩在时间$T$处消失。在第二种情况下，奇异性的轮廓描述的{em奇异} Yamabe收缩稍早$T$和匹配的{em Yamabe膨胀}稍后$T$。
This work addresses the {em singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have {em cylindrical behavior at infinity}. Their singularity profiles happen to be {em Yamabe solitons}, which are {em self-similar solutions} to the fast diffusion equation satisfied by the conformal factor of the evolving metric. The self-similar profile is determined by the second order asymptotics at infinity of the initial data which is matched with that of the corresponding self-similar solution. Solutions may become extinct at the extinction time $T$ of the cylindrical tail or may live longer than $T$. In the first case the singularity profile is described by a {em Yamabe shrinker} that becomes extinct at time $T$. In the second case, the singularity profile is described by a {em singular} Yamabe shrinker slightly before $T$ and by a matching {em Yamabe expander} slightly after $T$ .