Lagrangian mean curvature flow in pseudo-Euclidean space

Rongli Huang

Chinese Annals of Mathematics, Series B ›› 2011, Vol. 32 ›› Issue (2) : 187 -200.

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Chinese Annals of Mathematics, Series B ›› 2011, Vol. 32 ›› Issue (2) : 187 -200. DOI: 10.1007/s11401-011-0639-2
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Lagrangian mean curvature flow in pseudo-Euclidean space

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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.

Keywords

Indefinite metric / Self-expanding solution / Interior Schauder estimates / Logarithmic Monge-Ampère flow

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Rongli Huang. Lagrangian mean curvature flow in pseudo-Euclidean space. Chinese Annals of Mathematics, Series B, 2011, 32(2): 187-200 DOI:10.1007/s11401-011-0639-2

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References

[1]

Andrews B.. Pinching estimates and motions of hypersurfaces by curvature functions. J. Rein. Angew. Math., 2007, 608: 17-33

[2]

Caffarelli L., Nirenberg L., Spruck J.. The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian. Acta. Math., 1985, 155: 261-301

[3]

Caffarelli L.. Interior W 2.1 estimates for solutions of the Monge-Ampère equation. Ann. Math., 1990, 131(2): 135-150

[4]

Caffarelli L.. A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. Math., 1990, 131(1–2): 129-134

[5]

Chau, A., Chen, J. Y. and He, W. Y., Lagrangian Mean Curvature flow for entire Lipschitz graphs. arXiv:0902.3300

[6]

Chau, A., Chen, J. Y. and He, W. Y., Entire self-similar solutions to Lagrangian Mean curvature flow. arXiv: 0905.3869

[7]

Chen J. Y., Li J. Y.. Mean curvature flow of surface in 4-manifolds. Adv. Math., 2001, 163: 287-309

[8]

Chen J. Y., Li J. Y.. Singularity of mean curvature flow of Lagrangian submanifolds. Invent. Math., 2004, 156: 25-51

[9]

Colding, T. H. and Minicozzi, W. P., Generic mean curvature flow I: generic singularities. arXiv: 0908.3788

[10]

Gilbarg D., Trudinger N.. Elliptic Partial Differential Equations of Second Order, 1998 2nd ed. Berlin: Springer-Verlag

[11]

Han X. L., Li J. Y.. The mean curvature flow approach to the symplectic isotopy problem. Int. Math. Res. Not., 2005, 26: 1611-1620

[12]

Huang, R. L. and Bao, J. G., The blow up analysis of the general curve shortening flow. arXiv: 0908.2036

[13]

Huang, R. L. and Wang, Z. Z., On the entire self-shrinking solutions to Lagrangian mean curvature flow, Cal. Var. PDE, to appear.

[14]

Lieberman G. M.. Second Order Parabolic Differential Equations, 1996, Singapore: World Scientific

[15]

Lions P. L., Musiela M.. Convexity of solutions of parabolic equations. C. R. Acad. Sci. Paris, Ser. I, 2006, 342: 915-921

[16]

Neves A.. Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. Math., 2007, 168: 449-484

[17]

Pogorelov A. V.. On the improper convex affine hyperspheres. Geom Dedi., 1972, 1: 33-46

[18]

Protter M. H., Weinberger H. F.. Maximum Principle in Differential Equations, 1967, New Jersey: Prentice Hall

[19]

Smoczyk K.. Longtime existence of the Lagrangian mean curvature flow. Cal. Var. PDE, 2004, 20: 25-46

[20]

Smoczyk K., Wang M. T.. Mean curvature flows of Lagrangian submanifolds with convex potentials. J. Diff. Geom., 2002, 62: 243-257

[21]

Tso K. S.. On a real Monge-Ampere functional. Invent. Math., 1990, 101: 425-448

[22]

Xin Y. L.. Mean curvature flow with convex Gauss image. Chin. Ann. Math., 2008, 29B(2): 121-134

[23]

Xin, Y. L., Mean curvature flow with bounded Gauss image, preprint.

[24]

Xin Y. L.. Minimal Submanifolds and Related Topics, 2003, Singapore: World Scientific

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