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
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
must be a quadratic polynomial, where λ i are the eigenvalues of the Hessian D 2 u.
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Huang, R., Wang, Z. On the entire self-shrinking solutions to Lagrangian mean curvature flow. Calc. Var. 41, 321–339 (2011). https://doi.org/10.1007/s00526-010-0364-9
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DOI: https://doi.org/10.1007/s00526-010-0364-9