Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows

We prove that all entire smooth strictly convex self-shrinking solutions on ${\mathbb{R}}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we show that our argument works for a larger class of equations. In particular, we obtain rigidity results for entire self-shrinking solutions on ${\mathbb{C}}^n$ to the K"{a}hler-Ricci flow under certain conditions.