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

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

2009-11-15

arxiv.org/abs/0911.2849v7

Calc. Var. (2011) 41:321--339

Mathematics

Analysis of PDEs

Title changed, introduction rewritten and Theorem 1.3 added

Scientific paper

10.1007/s00526-010-0364-9

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 $D^{2}u$.

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