Abstract. We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regularity of Lagrangian mean curvature flow.Given two transverse Lagrangian planes R n in C n with sum of characteristic angles less than π, we show there exists a Lagrangian selfexpander asymptotic to this pair of planes. The Maslov class of these self-expanders is zero. Thus they can serve as local models for surgeries on Lagrangian mean curvature flow. Families of self-shrinkers and self-expanders with different topologies are also constructed. This paper generalizes the work of Anciaux [1], Joyce [12], Lawlor [15] and Lee and Wang [18,19].

Abstract. In an upcoming paper by Lee and Wang, we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow -a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimensions. We construct new higher-dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfson cones. Hamiltonian stationary self-shrinkers and self-expanders that are asymptotic to these Hamiltonian stationary cones are constructed as well. They can also be glued together to produce eternal solutions of the Brakke flow without mass loss. Finally, we show that the same conclusion holds for those Lagrangian self-similar examples recently found by Joyce, Tsui and the first author.

A Calabi-Yau manifold is a Kähler manifold with trivial canonical line bundle. It is proved by S.T. Yau [24] that in a Calabi-Yau manifold there exists a unique Ricci flat metric in its Kähler class. Therefore, we have two special forms ω and Ω in an n-dimensional Calabi-Yau manifold N , where ω is the Kähler form of the Ricci flat metric g and Ω is a parallel holomorphic (n, 0) form of unit length with respect to g.

In this paper, we generalize Colding and Minicozzi's work [5] on the stability of self-shrinkers in the hypersurface case to higher codimensional cases. The first and second variation formulae of the F -functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in [2] are unstable.