Complete Shrinking General Ricci Flow Soliton Systems

In this paper, we show that a complete shrinking general Ricci flow soliton system $(M,g,H,X,u,\lambda)$ with condition $h\geq0$ is compact if and only if $||X|| $ is bounded on $M$, where $h$ is the 2-form with components $h_{ij}=\frac{1}{2}H_{ikl}H_{j}^{kl}$. We also prove that a complete shrinking general Ricci flow system soliton has finite fundamental group.