Hamiltonian stationary cones and self-similar solutions in higher dimension

Yng-Ing Lee Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10047

Transactions of the American Mathematical Society, 362, (3), 1491-1503, 2010
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.
Hamiltonian stationary, Lagrangian mean curvature flow, selfshrinker, self-expander, eternal solution, Brakke flow
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@inproceedings{yng-ing2010hamiltonian,
  title={Hamiltonian stationary cones and self-similar solutions in higher dimension},
  author={Yng-Ing Lee, and Mu-Tao Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160821002228184130351},
  booktitle={Transactions of the American Mathematical Society},
  volume={362},
  number={3},
  pages={1491-1503},
  year={2010},
}
Yng-Ing Lee, and Mu-Tao Wang. Hamiltonian stationary cones and self-similar solutions in higher dimension. 2010. Vol. 362. In Transactions of the American Mathematical Society. pp.1491-1503. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160821002228184130351.
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