Abstract
The author establishes the long-time existence and convergence results of the mean curvature flow of entire Lagrangian graphs in the pseudo-Euclidean space, which is related to the logarithmic Monge-Ampère flow.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andrews, B., Pinching estimates and motions of hypersurfaces by curvature functions, J. Rein. Angew. Math., 608, 2007, 17–33.
Caffarelli, L., Nirenberg, L. and Spruck, J., The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta. Math., 155, 1985, 261–301.
Caffarelli, L., Interior W 2.1 estimates for solutions of the Monge-Ampère equation, Ann. Math., 131(2), 1990, 135–150.
Caffarelli, L., A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. Math., 131(1–2), 1990, 129–134.
Chau, A., Chen, J. Y. and He, W. Y., Lagrangian Mean Curvature flow for entire Lipschitz graphs. arXiv:0902.3300
Chau, A., Chen, J. Y. and He, W. Y., Entire self-similar solutions to Lagrangian Mean curvature flow. arXiv: 0905.3869
Chen, J. Y. and Li, J. Y., Mean curvature flow of surface in 4-manifolds, Adv. Math., 163, 2001, 287–309.
Chen, J. Y. and Li, J. Y., Singularity of mean curvature flow of Lagrangian submanifolds, Invent. Math., 156, 2004, 25–51.
Colding, T. H. and Minicozzi, W. P., Generic mean curvature flow I: generic singularities. arXiv: 0908.3788
Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1998.
Han, X. L. and Li, J. Y., The mean curvature flow approach to the symplectic isotopy problem, Int. Math. Res. Not., 26, 2005, 1611–1620.
Huang, R. L. and Bao, J. G., The blow up analysis of the general curve shortening flow. arXiv: 0908.2036
Huang, R. L. and Wang, Z. Z., On the entire self-shrinking solutions to Lagrangian mean curvature flow, Cal. Var. PDE, to appear.
Lieberman, G. M., Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.
Lions, P. L. and Musiela, M., Convexity of solutions of parabolic equations, C. R. Acad. Sci. Paris, Ser. I, 342, 2006, 915–921.
Neves, A., Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math., 168, 2007, 449–484.
Pogorelov, A. V., On the improper convex affine hyperspheres, Geom Dedi., 1, 1972, 33–46.
Protter, M. H. and Weinberger, H. F., Maximum Principle in Differential Equations, Prentice Hall, New Jersey, 1967.
Smoczyk, K., Longtime existence of the Lagrangian mean curvature flow, Cal. Var. PDE, 20, 2004, 25–46.
Smoczyk, K. and Wang, M. T., Mean curvature flows of Lagrangian submanifolds with convex potentials, J. Diff. Geom., 62, 2002, 243–257.
Tso, K. S., On a real Monge-Ampere functional, Invent. Math., 101, 1990, 425–448.
Xin, Y. L., Mean curvature flow with convex Gauss image, Chin. Ann. Math., 29B(2), 2008, 121–134.
Xin, Y. L., Mean curvature flow with bounded Gauss image, preprint.
Xin, Y. L., Minimal Submanifolds and Related Topics, World Scientific, Singapore, 2003.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, R. Lagrangian mean curvature flow in pseudo-Euclidean space. Chin. Ann. Math. Ser. B 32, 187–200 (2011). https://doi.org/10.1007/s11401-011-0639-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-011-0639-2