Lagrangian mean curvature flow for entire Lipschitz graphs

By Albert Chau, Jingyi Chen, Weiyong He. 2009

52 citations recorded by the source corpus.

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 $${{\mathbb R}^{2n}}$$ , we show that the parabolic Eq. 1.1 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 (1.2) 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 → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L∞. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cc024973141dddab3d73f51b771db66ad3bfb6d3

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In Applied Mathematics

1. Knut Smoczyk
2. D. Joyce
3. Y. Xin
4. Qi Ding

In Geometry and Topology

1. Knut Smoczyk
2. D. Joyce
3. Y. Xin
4. Qi Ding

Citations in the corpus

• Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, result): “A similar result holds for non-compact graphs [13].”