HAMILTONIAN STATIONARY SHRINKERS AND EXPANDERS FOR LAGRANGIAN MEAN CURVATURE FLOWS

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson in [SWO]. The Schoen Wolfson cones C(p,q) are obstructions to the existence problems of special Lagrangians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth oriented Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a weak formulation of the mean curvature flow. For any coprime pair (p, q) with p > q > 1, we construct such a solution that resolves one single Schoen-Wolfson cone C(p,q). Note that C(p,q) is stable only if p - q = 1. It thus provides an evidence to Schoen-Wolfson's conjecture that the (2, 1) cone is the only area-minimizing cone. Higher dimensional generalizations are also obtained.