The Ricci flow under almost non-negative curvature conditions
The Ricci flow under almost non-negative curvature conditions
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DOI：
10.1007/s00222-019-00864-7
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发表时间：
2017-07
影响因子：
3.1
通讯作者：
R. Bamler;Esther Cabezas-Rivas;Burkhard Wilking
中科院分区：
文献类型：
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作者：
R. Bamler;Esther Cabezas-Rivas;Burkhard Wilking
我们推广了大多数已知的Ricci流不变的非负曲率条件，限制较少的负界，保持足够的控制很短的时间。作为本文内容的一个例证，我们证明了曲率算子的特征值大于的度量可以在一定时间内由Ricci流演化，使得曲率算子的特征值保持大于.这里存在的时间和常数C只取决于维数和非连续性的程度。我们得到类似的推广其他不变曲率条件，包括积极的双全纯曲率的凯勒的情况下。我们还得到了主要定理的局部形式。作为我们几乎保持结果的应用，我们推导出了各种独立感兴趣的间隙和光滑结果，包括具有几乎非负曲率算子的非塌陷流形的分类和来自具有曲率下界的流形序列的奇异空间的光滑结果.我们还得到了几乎非负曲率开流形上Ricci流的短时存在性结果（不需要曲率上界）。
We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater thancan be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than. Here the time of existence and the constantConly depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kähler case. We also get a local version of the main theorem. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).