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

Abstract

The authors prove that the logarithmic Monge-Ampère 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}\arctanλ_{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 $D^{2}u$.

Title changed, introduction rewritten and Theorem 1.3 added

References in corpus (2)

• A Bernstein problem for special Lagrangian equations
• Mean curvature flow of monotone Lagrangian submanifolds