Lagrangian Mean Curvature flow for entire Lipschitz graphs
| dc.creator | Chau, Albert | |
| dc.creator | Chen, Jingyi | |
| dc.creator | He, Weiyong | |
| dc.date | 2009-02-19 | |
| dc.date.accessioned | 2026-07-07T12:44:07Z | |
| dc.date.available | 2026-07-07T12:44:07Z | |
| dc.description | We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\to \infty$. Our assumption on the Lipschitz norm is equivalent to the assumption that the underlying Lagrangian potential $u$ is uniformly convex with its Hessian bounded in $L^\infty$. We apply this result to prove a Bernstein type theorem for translating solitons, namely that if such an entire Lagrangian graph is a smooth translating soliton, then it must be a flat plane. We also prove convergence of the evolving graphs under additional conditions. | |
| dc.description | 22 pages | |
| dc.identifier | https://arxiv.org/abs/0902.3300 | |
| dc.identifier | http://arxiv.org/abs/0902.3300 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/220661 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53C44; 53A10 | |
| dc.title | Lagrangian Mean Curvature flow for entire Lipschitz graphs | |
| dc.type | text |