Abstract
The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space \({\mathbb {R}}^4\) with constant squared norm of the second fundamental form.
Similar content being viewed by others
References
Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23, 175–196 (1986)
Anciaux, H.: Construction of Lagrangian self-similar solutions to the mean curvature flow in \({\mathbb{C}}^n\). Geom. Dedic. 120, 37–48 (2006)
Brendle, S.: Embedded self-similar shrinkers of genus 0. Ann. Math. 183, 715–728 (2016)
Cao, H.-D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Part. Differ. Equ. 46, 879–889 (2013)
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)
Cheng, Q.-M., Ogata, S.: 2-Dimensional complete self-shrinkers in \({\mathbb{R}}^3\). Math. Z. 284, 537–542 (2016)
Cheng, Q.-M., Peng, Y.: Complete self-shrinkers of the mean curvature flow. Calc. Var. Part. Differ. Equ. 52, 497–506 (2015). https://doi.org/10.1007/s00526-014-0720-2
Cheng, Q.-M., Wei, G.: A gap theorem for self-shrinkers. Trans. Am. Math. Soc. 367, 4895–4915 (2015)
Cheng, X., Zhou, D.: Volume estimate about shrinkers. Proc. Am. Math. Soc. 141, 687–696 (2013)
Colding, T.H., Minicozzi, W.P., II.: Generic mean curvature flow I; Generic singularities. Ann. Math. 175, 755–833 (2012)
Ding, Q., Xin, Y.L.: Volume growth, eigenvalue and compactness for self-shrinkers. Asia J. Math. 17, 443–456 (2013)
Ding, Q., Xin, Y.L.: The rigidity theorems of self shrinkers. Trans. Am. Math. Soc. 366, 5067–5085 (2014)
Halldorsson, H.: Self-similar solutions to the curve shortening flow. Trans. Am. Math. Soc. 364, 5285–5309 (2012)
Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA: Proc. Sympos. Pure Math., 54, Part 1, Am. Math. Soc. Providence, R I 1993, 175–191 (1990)
Huisken, G.: Flow by mean curvature convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)
Kleene, S., Møller, N.M.: Self-shrinkers with a rotation symmetry. Trans. Am. Math. Soc. 366, 3943–3963 (2014)
Lawson, H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. 89, 187–197 (1969)
Le, N.Q., Sesum, N.: Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Commun. Anal. Geom. 19, 633–659 (2011)
Lee, Y.-I., Wang, M.-T.: Hamiltonian stationary cones and self-similar solutions in higher dimension. Trans. Am. Math. Soc. 362, 1491–1503 (2010)
Li, H., Wang, X.F.: New characterizations of the Clifford torus as a Lagrangian self-shrinkers. J. Geom. Anal. 27, 1393–1412 (2017)
Li, H., Wei, Y.: Classification and rigidity of self-shrinkers in the mean curvature flow. J. Math. Soc. Jpn. 66, 709–734 (2014)
Neves, A.: Singularities of Lagrangian mean curvature flow: monotone case. Math. Res. Lett. 17, 109–126 (2010)
Neves, A.: Recent progress on singularities of Lagrangian mean curvature flow. In: Surveys in geometric analysis and relativity, pp.413–438, Advanced Lectures in Mathematics (ALM), 20(2011), International Press, Somerville
Smoczyk, K.: The Lagrangian mean curvature flow, vol. 102 S. Univ. Leipzig (Habil), Leipzig (2000)
Smoczyk, K.: Self-shrinkers of the mean curvature flow in arbitrary co-dimension. Int. Math. Res. Not. 48, 2983–3004 (2005)
Strominger, A., Yau, S.T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243–259 (1996)
Yau, S.T.: Submanifolds with constant mean curvature I. Am. J. Math. 96, 346–366 (1974)
Acknowledgements
The authors would like to thank the referee for his/her valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Yuan-Long Xin for his 75th birthday.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The Qing-Ming Cheng was partially supported by JSPS Grant-in-Aid for Scientific Research (B): No. 16H03937. The Guoxin Wei was partly supported by NSFC Grant Nos. 12171164, 11771154, Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2018), Guangdong Natural Science Foundation Grant No. 2019A1515011451.
Rights and permissions
About this article
Cite this article
Cheng, QM., Hori, H. & Wei, G. Complete Lagrangian self-shrinkers in \({\mathbf {R}}^4\). Math. Z. 301, 3417–3468 (2022). https://doi.org/10.1007/s00209-022-03027-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-022-03027-2