Abstract
In this paper, we first introduce the concept of \(\xi \)-submanifold which is a natural generalization of self-shrinkers for the mean curvature flow and also an extension of \(\lambda \)-hypersurfaces to the higher codimension. Then, as the main result, we prove a rigidity theorem for Lagrangian \(\xi \)-submanifold in the complex 2-plane \({\mathbb C}^2\).
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References
Anciaux, H.: Construction of Lagrangian self-similar solutions to the mean curvature flow in \({\mathbb{C}}^{n}\). Geom. Dedic. 120, 37–48 (2006)
Arsie, A.: Maslov class and minimality in Calabi–Yau manifolds. J. Geom. Phys. 35, 145–156 (2000)
Castro, I., Lerma, A.M.: Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in the complex Euclidean plane. Proc. Am. Math. Soc. 138, 1821–1832 (2010)
Castro, I., Lerma, A.M.: The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow. Int. Math. Res. Not. 6, 1515–1527 (2014)
Cao, H.-D., Li, H.Z.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Partial Diff. Equ. 46, 879–889 (2013)
Cheng, Q.-M., Ogata, S., Wei, G.: Rigidity theorems of \(\lambda \)-hypersurfaces. arXiv arXiv:1403.4123v2 [math. DG], 1 Apr (2014)
Cheng, Q.-M., Wei, G.: Complete \(\lambda \)-hypersurfaces of weighted volume-preserving mean curvature flow. arXiv:1403.3177v3 [math. DG], 10 Jun (2014)
Cheng, Q.-M., Wei, G.: The Gauss image of \(\lambda \)-hypersurfaces and a Bernstein type problem. arXiv:1410.5302v1 [math.DG], 20 Oct (2014)
Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I; generic singularities. Ann. Math. 175, 755–833 (2012)
Gilbarg, D., Trudinger, N.S.: Elleptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Guang, Q.: Gap and rigidity theorems of \(\lambda \)-hypersurfaces. arXiv:1405.4871v1 [math. DG], 19 May (2014)
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)
Li, H. Z., Wang, X.F.: A new characterization of the clifford torus as a Lagrangian self-shrinker. arXiv:1505.05593v1 [math.DG], 21 May (2015)
Li, H.Z., Wei, Y.: Classification and rigidity of self-shrinkers in the mean curvature flow. J. Math. Soc. Jpn. 66, 709–734 (2014)
Li, X.X., Chang, X.F.: Rigidity theorems on the space-like \(\lambda \)-hypersurfaces in the Lorentzian space \({\mathbb{R}}^{n+1}_1\), preprint
Morvan, J.-M.: Classe de Maslov d’une immersion lagrangienne et minimalité. C. R. Acad. Sci. Paris I 292, 633–636 (1981)
Smoczyk, K.: Self-shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 48, 2983–3004 (2005)
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The authors really appreciate the kind suggestions and comments by the referee.
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Research supported by National Natural Science Foundation of China (Nos. 11171091, 11371018).
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Li, X., Chang, X. A rigidity theorem of \(\xi \)-submanifolds in \({\mathbb {C}}^{2}\) . Geom Dedicata 185, 155–169 (2016). https://doi.org/10.1007/s10711-016-0173-1
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DOI: https://doi.org/10.1007/s10711-016-0173-1