Hamiltonian stationary cones and self-similar solutions in higher dimension

TL;DR

This paper extends the construction of Hamiltonian stationary cones and self-similar solutions in higher dimensions, demonstrating their role in producing mass-preserving solutions to the mean curvature flow.

Contribution

It generalizes previous two-dimensional results to higher dimensions, introducing new Hamiltonian stationary cones and self-similar solutions with applications to Brakke flow.

Findings

Constructed higher-dimensional Hamiltonian stationary cones with various topologies.

Produced new self-shrinkers and self-expanders asymptotic to these cones.

Established solutions of Brakke flow with no mass loss.

Abstract

In [LW], 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 dimension. 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 also constructed. They can also be glued together to produce eternal solutions of the Brakke flow without mass loss. Finally, we show the same conclusion holds for those Lagrangian self-similar examples recently found by Joyce, Tsui and the first author in [JLT].