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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 Cp,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 Cp,q. Note that Cp,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.

All continuous GL(n) covariant valuations that map convex bodies to convex bodies are completely classified. This establishes a characterization of moment and projection operators and shows that the Holmes-Thompson area is the unique Minkowski area that is also a bivaluation.

Continuing the program of [DS] and [U1], we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Taubes invariant is easy to define, our invariants agree with those analogues. We also prove a vanishing result for some of the invariants that count nodal curves.

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration with smooth central fiber.