Let ( M , g ) be a closed Riemannian manifold. The RG-2 flow is defined by $$\begin{aligned} \frac{\partial }{\partial t} \, g(t) \, =\, -2 \mathrm {Ric}(t) \, -\, \frac{\alpha }{2} \mathrm {Rm}^2(t), \end{aligned}$$ ∂ ∂ t g ( t ) = - 2 Ric ( t ) - α 2 Rm 2 ( t ) , where $$ g = \mathrm {Riemannian \ metric}, \mathrm {Ric} = \mathrm {Ricci \ curvature, } \ \mathrm {Rm}^2_{ij}:=\mathrm {R}_{irmk}\mathrm {R}_j^{rmk},$$ g = Riemannian metric , Ric = Ricci curvature , Rm ij 2 : = R irmk R j rmk , and $$\alpha \ge 0$$ α ≥ 0 is a parameter. This is the geometric flow associated with the second-order approximation to the perturbative renormalization group flow for the nonlinear sigma model. It is invariant under diffeomorphisms, but not under scaling of the metric, the latter of which gives rise to several delicate problems from the point of view of geometric analysis. To address the lack of scaling we introduce a geometrically defined coupling constant $$\alpha _g$$ α g that leads to an equivalent, scale-invariant flow. We further find a modified Perelman entropy for the flow, and prove local existence of the resulting variational system. The crucial idea is to modify the flow by two diffeomorphisms, the first being the usual DeTurck diffeomorphism and the second being strictly related to the geometrical characterization of the coupling constant $$\alpha _g$$ α g . We minimize the entropy functional so introduced to characterize a natural extension $$\Lambda [g]$$ Λ [ g ] of the Perelman’s $$\lambda (g)$$ λ ( g ) –functional, and show that $$\Lambda [g]$$ Λ [ g ] is monotonic under the RG-2 flow. Although the modified Perelman entropy is monotonic, the RG-2 flow is not a gradient flow with respect this functional. We discuss this issue in detail, showing how to deform the functional in order to give rise to a gradient flow for a DeTurck modified RG-2 flow.

中文翻译：

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RG-2 流的缩放和熵

令 ( M , g ) 是一个封闭的黎曼流形。RG-2 流定义为 $$\begin{aligned} \frac{\partial }{\partial t} \, g(t) \, =\, -2 \mathrm {Ric}(t) \, - \, \frac{\alpha }{2} \mathrm {Rm}^2(t), \end{aligned}$$ ∂ ∂ tg ( t ) = - 2 Ric ( t ) - α 2 Rm 2 ( t ) , 其中 $$ g = \mathrm {Riemannian \ metric}, \mathrm {Ric} = \mathrm {Ricci \曲率, } \ \mathrm {Rm}^2_{ij}:=\mathrm {R}_{irmk} \mathrm {R}_j^{rmk},$$ g = 黎曼度量，Ric = Ricci 曲率，Rm ij 2 := R irmk R j rmk ，$$\alpha \ge 0$$ α ≥ 0 是一个参数. 这是与非线性 sigma 模型的微扰重整化群流的二阶近似相关的几何流。它在微分同胚下是不变的，但在度量的标度下不是不变的，从几何分析的角度来看，后者引起了几个微妙的问题。为了解决缺乏缩放的问题，我们引入了一个几何定义的耦合常数 $$\alpha _g$$ α g ，它导致等效的、尺度不变的流。我们进一步找到了流动的修正佩雷尔曼熵，并证明了所得变分系统的局部存在。关键的想法是通过两个微分同胚修改流，第一个是通常的 DeTurck 微分同胚，第二个与耦合常数 $$\alpha _g$$ α g 的几何特征严格相关。我们最小化如此引入的熵泛函来表征 Perelman 的 $$\lambda (g)$$ λ ( g ) – 泛函的自然扩展 $$\Lambda [g]$$ Λ [ g ]，并证明 $$\ Lambda [g]$$ Λ [ g ] 在 RG-2 流下是单调的。尽管修正的 Perelman 熵是单调的，但 RG-2 流就该函数而言不是梯度流。我们详细讨论了这个问题，展示了如何使函数变形，以便为 DeTurck 修改后的 RG-2 流产生梯度流。