It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.
References
S J Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom. 34 (1991) 491–514 MR1131441 0754.53006 10.4310/jdg/1214447218 euclid.jdg/1214447218 S J Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom. 34 (1991) 491–514 MR1131441 0754.53006 10.4310/jdg/1214447218 euclid.jdg/1214447218
H Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature flow in $\mathbb C^n$, Geom. Dedicata 120 (2006) 37–48 MR2252892 10.1007/s10711-006-9082-z H Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature flow in $\mathbb C^n$, Geom. Dedicata 120 (2006) 37–48 MR2252892 10.1007/s10711-006-9082-z
S Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991) 601–633 MR1100205 0731.53002 10.4310/jdg/1214446558 euclid.jdg/1214446558 S Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991) 601–633 MR1100205 0731.53002 10.4310/jdg/1214446558 euclid.jdg/1214446558
B-Y Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997) 277–297 MR1447186 0877.53041 10.2748/tmj/1178225151 euclid.tmj/1178225151 B-Y Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997) 277–297 MR1447186 0877.53041 10.2748/tmj/1178225151 euclid.tmj/1178225151
J Chen, W He, A note on singular time of mean curvature flow, Math. Z. 266 (2010) 921–931 MR2729297 1201.53075 10.1007/s00209-009-0604-x J Chen, W He, A note on singular time of mean curvature flow, Math. Z. 266 (2010) 921–931 MR2729297 1201.53075 10.1007/s00209-009-0604-x
C G Evans, J D Lotay, F Schulze, Remarks on the self-shrinking Clifford torus, preprint (2018) 1802.01423 C G Evans, J D Lotay, F Schulze, Remarks on the self-shrinking Clifford torus, preprint (2018) 1802.01423
K Groh, Singular behavior of equivariant Lagrangian mean curvature flow, PhD thesis, Leibniz Universität Hannover (2007) Available at \setbox0\makeatletter\@url http://nbn-resolving.de/urn:nbn:de:gbv:089-5378074038 {\unhbox0 http://nbn-resolving.de/urn:nbn:de:gbv:089-5378074038 1245.53004 K Groh, Singular behavior of equivariant Lagrangian mean curvature flow, PhD thesis, Leibniz Universität Hannover (2007) Available at \setbox0\makeatletter\@url http://nbn-resolving.de/urn:nbn:de:gbv:089-5378074038 {\unhbox0 http://nbn-resolving.de/urn:nbn:de:gbv:089-5378074038 1245.53004
K Groh, M Schwarz, K Smoczyk, K Zehmisch, Mean curvature flow of monotone Lagrangian submanifolds, Math. Z. 257 (2007) 295–327 MR2324804 1144.53084 10.1007/s00209-007-0126-3 K Groh, M Schwarz, K Smoczyk, K Zehmisch, Mean curvature flow of monotone Lagrangian submanifolds, Math. Z. 257 (2007) 295–327 MR2324804 1144.53084 10.1007/s00209-007-0126-3
R S Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995) 215–226 MR1316556 0827.53006 10.4310/jdg/1214456010 euclid.jdg/1214456010 R S Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995) 215–226 MR1316556 0827.53006 10.4310/jdg/1214456010 euclid.jdg/1214456010
D Joyce, Y-I Lee, M-P Tsui, Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom. 84 (2010) 127–161 MR2629511 1206.53071 10.4310/jdg/1271271795 euclid.jdg/1271271795 D Joyce, Y-I Lee, M-P Tsui, Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom. 84 (2010) 127–161 MR2629511 1206.53071 10.4310/jdg/1271271795 euclid.jdg/1271271795
F Martín, A Savas-Halilaj, K Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015) 2853–2882 MR3412395 10.1007/s00526-015-0886-2 F Martín, A Savas-Halilaj, K Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015) 2853–2882 MR3412395 10.1007/s00526-015-0886-2
A Neves, Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math. 168 (2007) 449–484 MR2299559 1119.53052 10.1007/s00222-007-0036-3 A Neves, Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math. 168 (2007) 449–484 MR2299559 1119.53052 10.1007/s00222-007-0036-3
A Neves, G Tian, Translating solutions to Lagrangian mean curvature flow, Trans. Amer. Math. Soc. 365 (2013) 5655–5680 MR3091260 1282.53057 10.1090/S0002-9947-2013-05649-8 A Neves, G Tian, Translating solutions to Lagrangian mean curvature flow, Trans. Amer. Math. Soc. 365 (2013) 5655–5680 MR3091260 1282.53057 10.1090/S0002-9947-2013-05649-8
A Ros, F Urbano, Lagrangian submanifolds of $\mathbb{C}^n$ with conformal Maslov form and the Whitney sphere, J. Math. Soc. Japan 50 (1998) 203–226 MR1484619 10.2969/jmsj/05010203 euclid.jmsj/1225376795 A Ros, F Urbano, Lagrangian submanifolds of $\mathbb{C}^n$ with conformal Maslov form and the Whitney sphere, J. Math. Soc. Japan 50 (1998) 203–226 MR1484619 10.2969/jmsj/05010203 euclid.jmsj/1225376795
K Smoczyk, Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature, Calc. Var. Partial Differential Equations 4 (1996) 155–170 MR1379198 0921.53025 10.1007/BF01189952 K Smoczyk, Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature, Calc. Var. Partial Differential Equations 4 (1996) 155–170 MR1379198 0921.53025 10.1007/BF01189952
K Smoczyk, Local non-collapsing of volume for the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019) art. id. 20, 14 pages MR3890797 1406.53076 10.1007/s00526-018-1458-z K Smoczyk, Local non-collapsing of volume for the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019) art. id. 20, 14 pages MR3890797 1406.53076 10.1007/s00526-018-1458-z
C Viana, A note on the evolution of the Whitney sphere along mean curvature flow, preprint (2018) 1802.02108 C Viana, A note on the evolution of the Whitney sphere along mean curvature flow, preprint (2018) 1802.02108
Y L Xin, Translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015) 1995–2016 MR3396441 10.1007/s00526-015-0853-y Y L Xin, Translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015) 1995–2016 MR3396441 10.1007/s00526-015-0853-y