AI 中文总结
本文研究正欧拉示性数曲面上带度量挠率的适配Ricci流，证明球面上无非平凡孤子，给出不收敛例子，并在实射影平面上证明收敛，进而得到球面在反对称条件下的收敛性，并利用Łojasiewicz--Simon不等式证明接近稳定点时的收敛。
AI 中文摘要
我们研究了正欧拉示性数曲面上具有度量挠率的联络的适配Ricci流。首先，我们证明在二维球面上不存在该流的任何非平凡孤子，从而证实了Branding--Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121) 的一个猜想。我们给出了一个显式的挠率数据族，使得相应的全局解在$\mathbb{S}^2$上不收敛。尽管如此，我们提供了流收敛到稳定点的若干充分条件。我们首先证明归一化适配Ricci流在$\mathbb{RP}^2$上总是收敛，这完全回答了Branding和Kröncke论文中的一个问题。利用这一点，我们推导出当初始度量和挠率一形式是反对称时，流在$\mathbb{S}^2$上收敛。我们还证明了该流的Łojasiewicz--Simon梯度不等式，并利用它证明当解接近任意稳定点时，流收敛到稳定点。
英文摘要
We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the $2$-sphere thus confirming a conjecture of Branding--Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on $\mathbb{S}^2$. Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on $\mathbb{RP}^2$, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on $\mathbb{S}^2$ whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz--Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.
Comments23 pages. All comments welcome!
