乘积流形上的Einstein-Lambda流
The Einstein-Lambda flow on product manifolds
摘要：
考虑乘积型空间流形上具有正宇宙学常数的真空爱因斯坦流。在空间维至少为4的情况下，只要其中至少有一个因子或允许一个具有正爱因斯坦常数的黎曼爱因斯坦度规，我们就证明了连续族的超时空模型的存在性。此外，我们表明，这些家庭属于较大的连续家庭的模型，有两个完整的时间方向，即不重叠。作为补充，我们证明了只要没有因子具有正曲率，则乘积类中的任何模型都在一个时间方向上膨胀，而在另一个时间方向上坍缩。特别地，一个因子的正曲率是这类内的负折射的必要判据。最后，我们将我们的研究结果的Nariai解决方案在三个空间维度的不稳定性，并指出为什么类似的建设在该维度的lapsing模型失败。目前的研究结果意味着，存在不同类别的初始数据表现出根本不同类型的长时间行为下的爱因斯坦流的空间维度是严格大于3。此外，这一行为与空间拓扑有关，通过存在正曲率的黎曼爱因斯坦度量。
英文摘要：
We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form. In spatial dimension at least four we show the existence of continuous families of recollapsing models whenever at least one of the factors or admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that these families belong to larger continuous families with models that have two complete time directions, i.e. do not recollapse. Complementarily, we show that whenever no factor has positive curvature, then any model in the product class expands in one time direction and collapses in the other. In particular, positive curvature of one factor is a necessary criterion for recollapse within this class. Finally, we relate our results to the instability of the Nariai solution in three spatial dimensions and point out why a similar construction of recollapsing models in that dimension fails. The present results imply that there exist different classes of initial data which exhibit fundamentally different types of long-time behavior under the Einstein flow whenever the spatial dimension is strictly larger than three. Moreover, this behavior is related to the spatial topology through the existence of Riemannian Einstein metrics of positive curvature.
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