Rigidity of entire self-shrinking solutions to K\"ahler-Ricci flow with strictly real convex potential

Abstract

We prove that every entire self-shrinking solution on $\mathbb{C}^n$ to the K\"{a}hler-Ricci flow with strictly real convex potential must be quadratic. The very same argument also gives a pointwise proof for the rigidity of entire self-shrinking solutions to Lagrangian mean curvature flow in pseudo-Euclidean space obtained by Q. Ding and Y.L. Xin. Furthermore, we show that our argument works for a larger class of equations.