Hamiltonian stationary cones and selfsimilar solutions in higher dimension
Authors:
YngIng Lee and MuTao Wang
Journal:
Trans. Amer. Math. Soc. 362 (2010), 14911503
MSC (2000):
Primary 53C44, 53D12; Secondary 58J35
Published electronically:
July 17, 2009
MathSciNet review:
2563738
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Abstract: In an upcoming paper by Lee and Wang, we construct examples of twodimensional Hamiltonian stationary selfshrinkers and selfexpanders for Lagrangian mean curvature flows, which are asymptotic to the union of two SchoenWolfson cones. These selfshrinkers and selfexpanders 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 higherdimensional Hamiltonian stationary cones of different topology as generalizations of the SchoenWolfson cones. Hamiltonian stationary selfshrinkers and selfexpanders 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 selfsimilar examples recently found by Joyce, Tsui and the first author.
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Additional Information
YngIng 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
MuTao Wang
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
Email:
mtwang@math.columbia.edu
DOI:
http://dx.doi.org/10.1090/S0002994709047953
Keywords:
Hamiltonian stationary,
Lagrangian mean curvature flow,
selfshrinker,
selfexpander,
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 962628M002.
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
