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

$$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$

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

$$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$

must be a quadratic polynomial, where λ _i are the eigenvalues of the Hessian D ² u.