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

TL;DR

This paper proves that all entire smooth strictly convex self-shrinking solutions to Hessian quotient flows are quadratic, extending rigidity results to broader classes of equations including Kähler-Ricci flow solutions.

Contribution

It establishes a general rigidity theorem for entire convex self-shrinking solutions to Hessian quotient flows, including applications to Kähler-Ricci flow.

Findings

All such solutions are quadratic functions.

The method applies to a broader class of equations.

Rigidity results for solutions on a9^n under certain conditions.

Abstract

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.