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

TL;DR

This paper proves that all entire self-shrinking solutions with strictly real convex potential to the K"ahler-Ricci flow are quadratic, extending to related equations and flows, demonstrating a rigidity property.

Contribution

It establishes a rigidity result for entire self-shrinking solutions to the K"ahler-Ricci flow and related flows, showing they must be quadratic under certain conditions.

Findings

Self-shrinking solutions are quadratic under given conditions.

The proof applies to Lagrangian mean curvature flow in pseudo-Euclidean space.

The argument extends to a broader class of equations.

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.