On the second boundary value problem for Lagrangian mean curvature flow

Global Second Derivative Estimates for the Second Boundary Value Problem of the Prescribed Affine Mean Curvature and Abreu’s Equations

In this paper we prove the global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature equation where the affine mean curvature is only assumed to be in L. Our result extends previous result by Trudinger and Wang in the case of globally bounded affine mean curvature and also covers Abreu’s equation.

Global Second Derivative Estimates for the Second Boundary Value Problem of the Prescribed Affine Mean Curvature and Abreu’s Equations

Translating Solutions to Lagrangian Mean Curvature Flow

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an L bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.

Mean Curvature Flow and Lagrangian Embeddings

In this note we provide examples of compact embedded lagrangians in Cn for any n ≥ 2 that under mean curvature flow develop singularities in finite time. When n is odd the lagrangians can be taken to be orientable. By gluing these lagrangians onto a special lagrangian embedding L we provide examples of compact embedded lagrangians in a Calabi-Yau manifold that under mean curvature flow develop ...

Lagrangian Mean Curvature Flow for Entire Lipschitz Graphs

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, we show that the parabolic equation (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...