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 arbitrarily many singularities in finite time. These lagrangians look like the special lagrangian L with “filigree” attached at points. The construction of these examples has been motivated, at least in part, by the desire to obtain a better understanding of a recent conjecture of Thomas-Yau [T-Y]. Given a compact lagrangian submanifold in a CalabiYau manifold with zero Maslov index, Thomas and Yau conjecture that mean curvature flow exists for all time and converges to a smooth special lagrangian submanifold. (Actually to eliminate the possibility that in the limit the lagrangian degenerates into a connect sum, Thomas-Yau restrict the range of the “grading” on the lagrangian. For a more detailed discussion of the conjecture see below.) The examples we construct have a strikingly different behavior. However because they are not known to have zero Maslov index they are not counterexamples to the conjecture. On the other hand if they did have zero Maslov index then they would be counterexamples. The authors are grateful to Mu-Tao Wang for useful discussions on the topics of this note and to Richard Thomas for comments on an earlier version of this note.