更多关于Cotton Flow
More on Cotton Flow
摘要：
棉花流倾向于将三流形上的给定初始度量演化为共形平坦度量。在这里，我们阐述了棉花流的早期工作，并研究了它的线性化版本，它周围的一个通用的初始指标，采用修改后的形式的DeTurck技巧。我们表明，周围的平坦空间的流动，作为一个临界点，减少到各向异性推广的线性KdV方程与复杂的色散关系，其中之一是一个不稳定的模式，使平坦空间不稳定的小扰动。我们还证明了Einstein空间和某些共形平坦的非Einstein空间是线性不稳定的。我们改进的梯度流形式主义和计算的熵的第二变分，并表明，一般的临界点扩展的棉花孤子。我们研究了这些解的一些性质，并找到了一个由Cotton孤子和Ricci孤子构成的拓扑质量孤子。在Lorentzian签名中，我们还证明了pp波度规是Cotton孤子和Ricci孤子。
英文摘要：
Cotton flow tends to evolve a given initial metric on a three manifold to a conformally flat one. Here we expound upon the earlier work on Cotton flow and study the linearized version of it around a generic initial metric by employing a modified form of the DeTurck trick. We show that the flow around the flat space, as a critical point, reduces to an anisotropic generalization of linearized KdV equation with complex dispersion relations one of which is an unstable mode, rendering the flat space unstable under small perturbations. We also show that Einstein spaces and some conformally flat non-Einstein spaces are linearly unstable. We refine the gradient flow formalism and compute the second variation of the entropy and show that generic critical points are extended Cotton solitons. We study some properties of these solutions and find a Topologically Massive soliton that is built from Cotton and Ricci solitons. In the Lorentzian signature, we also show that the pp-wave metrics are both Cotton and Ricci solitons.
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