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Solitons arise either when singularities form under the mean curvature flow or when the solution exists for long times but the immersions do not necessarily converge to a minimal submanifold. The most important examples are self-similar shrinking and expanding solutions, as well as translating solitons.

According to a work by Strominger, Yau, and Zaslow, the moduli space of minimal Lagrangian tori in Calabi–Yau manifolds M is directly related to the mirror manifold of M. For this reason, the mean curvature flow is an important tool for understanding this connection. Key questions in this context include the existence of minimal Lagrangian tori and the compactness of the moduli space. In particular, in relation to these questions and the Thomas–Yau conjecture, the singularities that arise in the mean curvature flow of Lagrangian submanifolds still need to be better understood.

If z=(u,v)z=(u,v) is a regular curve in R² and Sdenotes the standard embedding of the (m−1)-dimensional unit sphere, then F(s,x):=(u(s)S(x), v(s)S(x))F(s,x):=(u(s)S(x),v(s)S(x)) defines an m-dimensional Lagrangian immersion. This immersion is invariant under the isometry group O(m); Lagrangian submanifolds of this type are called equivariant. The Whitney sphere (see the image on the left) is the best-known example of an equivariant Lagrangian submanifold. The mean curvature flow for equivariant submanifolds can be completely described by the evolution of the generating profile curves zz. While singularities in this special case are now very well understood, important open questions remain. For example, the nature of the singularity in cases where it forms at the origin is still not fully clarified.

In mean curvature flow, one generally distinguishes between two types of singularities, called Type I and Type II. Type I singularities produce self-similarly shrinking solutions and do not occur in Lagrangian mean curvature flow when the Maslov class is trivial. A Type II singularity, on the other hand, gives rise to a so-called eternal solution. This class includes, in particular, translating solutions such as the grim reaper. The moduli spaces of Type I and Type II singularities are still very poorly understood. For example, it is not known under which conditions Type II singularities automatically lead to translating solutions (note: for convex hypersurfaces there is a result on this due to Richard Hamilton).

A map f between Riemannian manifolds (M,g), (N,h) is called minimal if the graph Γ of f is a minimal submanifold of M×N with respect to the product metric. The flow therefore provides a means to better understand the existence of minimal maps between Riemannian manifolds. Under suitable assumptions on the Riemannian curvatures of M and N, significant recent results have been obtained for both the class of contractions and the class of area-decreasing maps.

If f is a symplectomorphism between Kähler–Einstein manifolds of the same scalar curvature, then mean curvature flow preserves this property. This provides another special case of the Lagrangian mean curvature flow. While the flow has been thoroughly studied for symplectomorphisms between Riemann surfaces, in higher dimensions many important questions remain open. For instance, one would like to understand under which conditions symplectomorphisms can be deformed into biholomorphisms.

A 1-form θ on a differentiable manifold M defines a graph in the cotangent bundle of M, and this graph is Lagrangian precisely when θ is closed. Since the cotangent bundle is in general not a Kähler–Einstein manifold, mean curvature flow does not preserve the Lagrangian condition in these cases. However, by suitably modifying the flow equation, it has been possible to construct a flow that preserves the closedness of the forms θ. For this modified mean curvature flow, very strong results are currently available for cotangent bundles of Riemannian manifolds with positive curvature. This is also of interest in the context of a variant of the Arnol'd conjecture.