Finite time singularities for Lagrangian mean curvature flow. (English) Zbl 1315.53074

Neves, André

Ann. Math. (2) 177, No. 3, 1029-1076 (2013).

The main result of the paper is as follows: let \(M\) be a Calabi-Yau manifold of real dimension 4 and \(\Sigma\) an embedded Lagrangian in \(M\). Then there exists a Lagrangian \(L\) which is Hamiltonian isotopic to \(\Sigma\) and so that the Lagrangian mean curvature flow with initial condition \(L\) has a finite time singularity. This disproves a conjecture of Mu-Tao Wang which is a weaker version of a conjecture of Thomas-Yau concerning long time existence (and convergence) of Lagrangian mean curvature flow satisfying a condition of stability.

Reviewer: Sylvain Maillot (Montpellier)

MSC:

53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53D12	Lagrangian submanifolds; Maslov index

Keywords:

special Lagrangian; Calabi-Yau; Hamiltonian isotopy; mean curvature flow

References:

[1]	H. Anciaux, ”Construction of Lagrangian self-similar solutions to the mean curvature flow in \(\mathbb C^n\),” Geom. Dedicata, vol. 120, pp. 37-48, 2006. · Zbl 1098.35074
[2]	S. Angenent, ”Parabolic equations for curves on surfaces. I. Curves with \(p\)-integrable curvature,” Ann. of Math., vol. 132, iss. 3, pp. 451-483, 1990. · Zbl 0789.58070
[3]	S. Angenent, ”Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions,” Ann. of Math., vol. 133, iss. 1, pp. 171-215, 1991. · Zbl 0749.58054
[4]	Y. Eliashberg and L. Polterovich, ”Local Lagrangian \(2\)-knots are trivial,” Ann. of Math., vol. 144, iss. 1, pp. 61-76, 1996. · Zbl 0872.57030
[5]	M. A. Grayson, ”Shortening embedded curves,” Ann. of Math., vol. 129, iss. 1, pp. 71-111, 1989. · Zbl 0686.53036
[6]	G. Huisken, ”Asymptotic behavior for singularities of the mean curvature flow,” J. Differential Geom., vol. 31, iss. 1, pp. 285-299, 1990. · Zbl 0694.53005
[7]	T. Ilmanen, ”Elliptic regularization and partial regularity for motion by mean curvature,” Mem. Amer. Math. Soc., vol. 108, iss. 520, p. x, 1994. · Zbl 0798.35066
[8]	T. Ilmanen, Singularities of Mean Curvature Flow of Surfaces. · Zbl 0759.53035
[9]	A. Neves, ”Singularities of Lagrangian mean curvature flow: zero-Maslov class case,” Invent. Math., vol. 168, iss. 3, pp. 449-484, 2007. · Zbl 1119.53052
[10]	A. Neves and G. Tian, Translating solutions to Lagrangian mean curvature flow. · Zbl 1282.53057
[11]	J. A. Oaks, ”Singularities and self intersections of curves evolving on surfaces,” Indiana Univ. Math. J., vol. 43, iss. 3, pp. 959-981, 1994. · Zbl 0835.53048
[12]	R. Schoen and J. Wolfson, ”Minimizing area among Lagrangian surfaces: the mapping problem,” J. Differential Geom., vol. 58, iss. 1, pp. 1-86, 2001. · Zbl 1052.53056
[13]	R. Schoen and J. Wolfson, Mean curvature flow and Lagrangian embeddings. · Zbl 1031.53112
[14]	L. Simon, Lectures on Geometric Measure Theory, Canberra: Australian National University Centre for Mathematical Analysis, 1983, vol. 3. · Zbl 0546.49019
[15]	K. Smoczyk, A canonical way to deform a Lagrangian submanifold.
[16]	R. P. Thomas and S. -T. Yau, ”Special Lagrangians, stable bundles and mean curvature flow,” Comm. Anal. Geom., vol. 10, iss. 5, pp. 1075-1113, 2002. · Zbl 1115.53054
[17]	M. Wang, ”Mean curvature flow of surfaces in Einstein four-manifolds,” J. Differential Geom., vol. 57, iss. 2, pp. 301-338, 2001. · Zbl 1035.53094
[18]	M. Wang, ”Some recent developments in Lagrangian mean curvature flows,” in Surveys in Differential Geometry. Vol. XII. Geometric Flows, Int. Press, Somerville, MA, 2008, vol. 12, pp. 333-347. · Zbl 1169.53052
[19]	B. White, ”A local regularity theorem for mean curvature flow,” Ann. of Math., vol. 161, iss. 3, pp. 1487-1519, 2005. · Zbl 1091.53045
[20]	J. Wolfson, ”Lagrangian homology classes without regular minimizers,” J. Differential Geom., vol. 71, iss. 2, pp. 307-313, 2005. · Zbl 1097.53052