Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: July 17, 2009
MathSciNet review: 2563738
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • 1. K.A. Brakke, The motion of a surface by its mean curvature. Mathematical Notes, Princeton University Press, 1978. MR 485012 (82c:49035)
  • 2. M. Feldman; T. Ilmanen; D. Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Differential Geom. 65 (2003), no. 2, 169-209. MR 2058261 (2005e:53102)
  • 3. R. Harvey; H.B. Lawson, Calibrated geometries. Acta Math. 148 (1982), 48-156. MR 666108 (85i:53058)
  • 4. M. Haskins, Special Lagrangian cones. Amer. J. Math. 126 (2004), no. 4, 845-871. MR 2075484 (2005e:53074)
  • 5. M. Haskins, The geometric complexity of special Lagrangian $ T\sp 2$-cones. Invent. Math. 157 (2004), no. 1, 11-70. MR 2135184 (2005m:53085)
  • 6. G. Huisken, Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom. 31 (1990), no. 1, 285-299. MR 1030675 (90m:53016)
  • 7. D. Joyce, Constructing special Lagrangian $ m$-folds in $ \mathbb{C}\sp m$ by evolving quadrics. Math. Ann. 320 (2001), no. 4, 757-797. MR 1857138 (2002j:53066)
  • 8. D. Joyce, Special Lagrangian $ m$-folds in $ \mathbb{C}\sp m$ with symmetries. Duke Math. J. 115 (2002), no. 1, 1-51. MR 1932324 (2003m:53083)
  • 9. D.D. Joyce; Y.I. Lee; M.P. Tsui, Self-similar solutions and translating solutions for Lagrangian mean curvature flow. arXiv: math.DG/0801.3721.
  • 10. Y.I. Lee; M.T. Wang, Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows. To appear in J. Differential Geom., also available at arXiv: math.DG/0707.0239.
  • 11. K. Smoczyk, Angle theorems for the Lagrangian mean curvature flow. Math. Z. 240 (2002), no. 4, 849-883. MR 1922733 (2003g:53120)
  • 12. K. Smoczyk; M.-T. Wang, Mean curvature flows of Lagrangian submanifolds with convex potentials. J. Differential Geom. 62 (2002), no. 2, 243-257. MR 1988504 (2004d:53086)
  • 13. R. Schoen; J.G. Wolfson Minimizing area among Lagrangian surfaces: The mapping problem. J. Differential Geom. 58 (2001), no. 1, 1-86. MR 1895348 (2003c:53119)
  • 14. A. Strominger; S.-T. Yau; E. Zaslow, Mirror symmetry is $ T$-duality. Nuclear Phys. B 479 (1996), no. 1-2, 243-259. MR 1429831 (97j:32022)
  • 15. M.-T. Wang, Deforming area preserving diffeomorphism of surfaces by mean curvature flow. Math. Res. Lett. 8 (2001), no. 5-6, 651-662. MR 1879809 (2003f:53122)
  • 16. M.-T. Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148 (2002), no. 3, 525-543. MR 1908059 (2003b:53073)
  • 17. J.G. Wolfson, Lagrangian homology classes without regular minimizers. J. Differential Geom. 71 (2005), 307-313. MR 2197143 (2006j:53119)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C44, 53D12, 58J35

Retrieve articles in all journals with MSC (2000): 53C44, 53D12, 58J35

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

Mu-Tao Wang
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

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

American Mathematical Society