Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in the complex Euclidean plane

Castro, Ildefonso; Lerma, Ana

doi:10.1090/S0002-9939-09-10134-X

Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in the complex Euclidean plane
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by Ildefonso Castro and Ana M. Lerma PDF

Proc. Amer. Math. Soc. 138 (2010), 1821-1832 Request permission

Abstract:

We classify all Hamiltonian stationary Lagrangian surfaces in the complex Euclidean plane which are self-similar solutions of the mean curvature flow.

References

Henri Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature flow in $\Bbb C^n$, Geom. Dedicata 120 (2006), 37–48. MR 2252892, DOI 10.1007/s10711-006-9082-z
H. Anciaux and I. Castro. Constructions of Hamiltonian-minimal Lagrangian submanifolds in complex Euclidean space. Preprint, 2009. arXiv:0906.4305 [math.DG]
Ildefonso Castro and Bang-Yen Chen, Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves, Tohoku Math. J. (2) 58 (2006), no. 4, 565–579. MR 2297200
Ildefonso Castro and Francisco Urbano, Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form, Tohoku Math. J. (2) 45 (1993), no. 4, 565–582. MR 1245723, DOI 10.2748/tmj/1178225850
Ildefonso Castro and Francisco Urbano, Examples of unstable Hamiltonian-minimal Lagrangian tori in $\mathbf C^2$, Compositio Math. 111 (1998), no. 1, 1–14. MR 1611051, DOI 10.1023/A:1000332524827
Jingyi Chen and Jiayu Li, Singularity of mean curvature flow of Lagrangian submanifolds, Invent. Math. 156 (2004), no. 1, 25–51. MR 2047657, DOI 10.1007/s00222-003-0332-5
K. Groh, M. Schwarz, K. Smoczyk, and K. Zehmisch, Mean curvature flow of monotone Lagrangian submanifolds, Math. Z. 257 (2007), no. 2, 295–327. MR 2324804, DOI 10.1007/s00209-007-0126-3
Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726
Frédéric Hélein and Pascal Romon, Hamiltonian stationary Lagrangian surfaces in $\Bbb C^2$, Comm. Anal. Geom. 10 (2002), no. 1, 79–126. MR 1894142, DOI 10.4310/CAG.2002.v10.n1.a5
Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
D. Joyce, Y.-I. Lee and M.-P. Tsui. Self-similar solutions and translating solutions for Lagrangian mean curvature flow. arXiv:0801.3721 [math.DG]
Y.-I. Lee and M.-T. Wang. Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows. in J. Differential Geom. 83 (2009), 27–42.
Y.-I. Lee and M.-T. Wang. Hamiltonian stationary cones and self-similar solutions in higher dimension. Trans. Amer. Math. Soc. 362 (2010) 1491–1503.
William P. Minicozzi II, The Willmore functional on Lagrangian tori: its relation to area and existence of smooth minimizers, J. Amer. Math. Soc. 8 (1995), no. 4, 761–791. MR 1311825, DOI 10.1090/S0894-0347-1995-1311825-9
André Neves, Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math. 168 (2007), no. 3, 449–484. MR 2299559, DOI 10.1007/s00222-007-0036-3
A. Neves. Singularities of Lagrangian mean curvature flow: monotone case. arXiv:math06068401v1 [math.DG], 2006.
Yong-Geun Oh, Volume minimization of Lagrangian submanifolds under Hamiltonian deformations, Math. Z. 212 (1993), no. 2, 175–192. MR 1202805, DOI 10.1007/BF02571651
K. Smoczyk. A canonical way to deform a Lagrangian submanifold. Preprint, dg-ga/9605005.
R. Schoen and J. Wolfson, Minimizing area among Lagrangian surfaces: the mapping problem, J. Differential Geom. 58 (2001), no. 1, 1–86. MR 1895348
Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831, DOI 10.1016/0550-3213(96)00434-8
R. P. Thomas and S.-T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113. MR 1957663, DOI 10.4310/CAG.2002.v10.n5.a8
Mu-Tao Wang, Mean curvature flow of surfaces in Einstein four-manifolds, J. Differential Geom. 57 (2001), no. 2, 301–338. MR 1879229