On the entire self-shrinking solutions to Lagrangian mean curvature flow

TL;DR

This paper proves that under certain conditions, entire convex solutions to specific Lagrangian mean curvature flow equations must be quadratic polynomials, extending understanding of self-shrinking solutions in geometric analysis.

Contribution

It establishes decay estimates for the logarithmic Monge-Ampère flow and characterizes entire solutions as quadratic polynomials under eigenvalue bounds, advancing the classification of self-shrinking solutions.

Findings

Solutions are quadratic polynomials under eigenvalue bounds.

Decay estimates for the Monge-Ampère flow are established.

Entire convex solutions are characterized explicitly.

Abstract

The authors prove that the logarithmic Monge-Amp\`{e}re flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time $t=0$. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation {equation*} \det D^{2}u=\exp\{n(-u+1/2\sum_{i=1}^{n}x_{i}\frac{\partial u}{\partial x_{i}})\}, {equation*} should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function $|x|^{2}D^{2}u$ at infinity has an uniform positive lower bound larger than $2(1-1/n)$. Using a similar method, we can prove that every classical convex or concave solution of the equation {equation*} \sum_{i=1}^{n}\arctan\lambda_{i}=-u+1/2\sum_{i=1}^{n}x_{i}\frac{\partial u}{\partial x_{i}}. {equation*} must be a quadratic polynomial, where $\lambda_{i}$ are the eigenvalues of the Hessian $D^{2}u$.