Hamiltonian stationary cones and self-similar solutions in higher dimension

Authors:
Yng-Ing Lee and Mu-Tao Wang

Journal:
Trans. Amer. Math. Soc. **362** (2010), 1491-1503

MSC (2000):
Primary 53C44, 53D12; Secondary 58J35

DOI:
https://doi.org/10.1090/S0002-9947-09-04795-3

Published electronically:
July 17, 2009

MathSciNet review:
2563738

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Abstract | References | Similar Articles | Additional Information

Abstract: 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.

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Additional Information

**Yng-Ing Lee**

Affiliation:
Department of Mathematics and Taida Institute for Mathematical Sciences, National Taiwan University, Taipei, Taiwan – and – National Center for Theoretical Sciences, Taipei Office, Old Mathematics Building, National Taiwan University, Taipei 10617, Taiwan

Email:
yilee@math.ntu.edu.tw

**Mu-Tao Wang**

Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027

Email:
mtwang@math.columbia.edu

DOI:
https://doi.org/10.1090/S0002-9947-09-04795-3

Keywords:
Hamiltonian stationary,
Lagrangian mean curvature flow,
self-shrinker,
self-expander,
eternal solution,
Brakke flow

Received by editor(s):
February 12, 2008

Published electronically:
July 17, 2009

Additional Notes:
The first author would like to thank R. Schoen for helpful discussions and hospitality during her visit at Stanford University. The first author was supported by Taiwan NSC grant 96-2628-M-002.

The second author wishes to thank the support of the Taida Institute for Mathematical Sciences during the preparation of this article. The second author was supported by NSF grant DMS0605115 and a Sloan research fellowship.

Article copyright:
© Copyright 2009
American Mathematical Society