Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 196751, 9 pages
http://dx.doi.org/10.1155/2014/196751
Research Article

Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007, China

Received 14 January 2014; Accepted 15 June 2014; Published 6 July 2014

Academic Editor: Ljubomir B. Ćirić

Copyright © 2014 Ruiwei Xu and Linfen Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. H. Colding and I. Minicozzi, “Generic mean curvature flow I: generic singularities,” Annals of Mathematics, vol. 175, no. 2, pp. 755–833, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. K. Ecker and G. Huisken, “Mean curvature evolution of entire graphs,” Annals of Mathematics, vol. 130, no. 3, pp. 453–471, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  3. G. Huisken, “Asymptotic behavior for singularities of the mean curvature flow,” Journal of Differential Geometry, vol. 31, no. 1, pp. 285–299, 1990. View at Google Scholar · View at MathSciNet
  4. G. Huisken, “Local and global behaviour of hypersurfaces moving by mean curvature,” in Differential Geometry: Partial Differential Equations on Manifolds, vol. 54 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1993. View at Google Scholar
  5. K. Smoczyk, “Self-shrinkers of the mean curvature flow in arbitrary codimension,” International Mathematics Research Notices, no. 48, pp. 2983–3004, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. L. Wang, “A Bernstein type theorem for self-similar shrinkers,” Geometriae Dedicata, vol. 151, pp. 297–303, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. H. D. Cao and H. Li, “A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension,” Calculus of Variations and Partial Differential Equations, vol. 46, no. 3-4, pp. 879–889, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. A. Chau, J. Chen, and Y. Yuan, “Rigidity of entire self-shrinking solutions to curvature flows,” Journal für die Reine und Angewandte Mathematik, vol. 664, pp. 229–239, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Q. Ding and Z. Z. Wang, On the self-shrinking systems in arbitrary codimension spaces, http://arxiv.org/abs/1012.0429.
  10. Q. Ding and Y. L. Xin, “The Rigidity theorems for Lagrangian self-shrinkers,” Journal für die Reine und Angewandte Mathematik, 2012. View at Publisher · View at Google Scholar
  11. K. Ecker, “Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space,” Journal of Differential Geometry, vol. 46, no. 3, pp. 481–498, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. R. Huang and Z. Wang, “On the entire self-shrinking solutions to Lagrangian mean curvature flow,” Calculus of Variations and Partial Differential Equations, vol. 41, no. 3-4, pp. 321–339, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. Y. L. Xin, “Mean curvature flow with bounded Gauss image,” Results in Mathematics, vol. 59, no. 3-4, pp. 415–436, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. Y. L. Xin, Minimal Submanifolds and Related Topics, World Scientific Publishing, 2003. View at MathSciNet
  15. A.-M. Li, R. Xu, U. Simon, and F. Jia, Affine Bernstein Problems and Monge-Ampère Equations, World Scientific, Singapore, 2010. View at MathSciNet
  16. A. M. Li and R. W. Xu, “A rigidity theorem for an affine Kähler-Ricci flat graph,” Results in Mathematics, vol. 56, no. 1–4, pp. 141–164, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. A.-M. Li and R. Xu, “A cubic form differential inequality with applications to affine Kähler-Ricci flat manifolds,” Results in Mathematics, vol. 54, no. 3-4, pp. 329–340, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. R. W. Xu and R. L. Huang, “On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space I,” Acta Mathematica Sinica (English Series), vol. 29, no. 7, pp. 1369–1380, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A. V. Pogorelov, The Minkowski Multidimensional Problem, John Wiley & Sons, New York, NY, USA, 1978.
  20. E. Calabi, “Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens,” The Michigan Mathematical Journal, vol. 5, pp. 105–126, 1958. View at Publisher · View at Google Scholar · View at MathSciNet