Optimal curvature estimates for homogeneous Ricci flows
Optimal curvature estimates for homogeneous Ricci flows
复制标题
DOI：
10.1093/imrn/rnx256
复制
发表时间：
2016-04
期刊：
影响因子：
--
通讯作者：
Christoph Bohm;Ramiro A. Lafuente;Miles Simon
中科院分区：
文献类型：
--
作者：
Christoph Bohm;Ramiro A. Lafuente;Miles Simon
证明了齐次Ricci流的一致曲率估计：对于定义在$[0，t]$上的解，t$时刻的曲率张量的范数有界于$C（n）/t$和$C（n）（scal（g（t））- scal（g（0）$的最大值.这是用来表明有限灭绝时间的解决方案是I型，不朽的解决方案是III型和古老的解决方案是I型，其中所有涉及的常数只取决于维数$n$。进一步的结果是，一个紧致齐次空间上的非坍缩齐次古解来自同一空间上的唯一爱因斯坦度量。利用齐性空间上Ricci平坦性的间隙定理证明了上述曲率估计。这个缺口定理的证明是矛盾的，并使用了局部W^{2，p}$收敛结果，该结果在没有对称性假设的情况下成立。
We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The above curvature estimates are proved using a gap theorem for Ricci-flatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local $W^{2,p}$ convergence result, which holds without symmetry assumptions.