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

The authors prove that the logarithmic Monge-AmpSre 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{n(-u + 1/2 Sigma(n)(i=1)x(i)partial derivative u/partial derivative x(i))}, 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 Sigma(n)(i=1)arctan lambda(i) = -u + 1/2 Sigma(n)(i=1)x(i)partial derivative u/partial derivative x(i) must be a quadratic polynomial, where lambda(i) are the eigenvalues of the Hessian D(2)u.