Geometry & Topology

Lagrangian mean curvature flow of Whitney spheres

Andreas Savas-Halilaj and Knut Smoczyk

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Geom. Topol., Volume 23, Number 2 (2019), 1057-1084.

Received: 4 April 2018
Accepted: 13 July 2018
First available in Project Euclid: 17 April 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Lagrangian mean curvature flow equivariant Lagrangian submanifolds type-II singularities


Savas-Halilaj, Andreas; Smoczyk, Knut. Lagrangian mean curvature flow of Whitney spheres. Geom. Topol. 23 (2019), no. 2, 1057--1084. doi:10.2140/gt.2019.23.1057.

Export citation


  • S J Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom. 34 (1991) 491–514
  • H Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature flow in $\mathbb C^n$, Geom. Dedicata 120 (2006) 37–48
  • S Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991) 601–633
  • V Borrelli, B-Y Chen, J-M Morvan, }, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 1485–1490
  • B-Y Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997) 277–297
  • J Chen, W He, A note on singular time of mean curvature flow, Math. Z. 266 (2010) 921–931
  • C G Evans, J D Lotay, F Schulze, Remarks on the self-shrinking Clifford torus, preprint (2018)
  • K Groh, Singular behavior of equivariant Lagrangian mean curvature flow, PhD thesis, Leibniz Universität Hannover (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • K Groh, M Schwarz, K Smoczyk, K Zehmisch, Mean curvature flow of monotone Lagrangian submanifolds, Math. Z. 257 (2007) 295–327
  • R S Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995) 215–226
  • 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
  • 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
  • A Neves, Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math. 168 (2007) 449–484
  • A Neves, G Tian, Translating solutions to Lagrangian mean curvature flow, Trans. Amer. Math. Soc. 365 (2013) 5655–5680
  • 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
  • K Smoczyk, Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature, Calc. Var. Partial Differential Equations 4 (1996) 155–170
  • K Smoczyk, Der Lagrangesche mittlere Krümmungsfluss, Habilitationsschrift, Universität Leipzig (2000)
  • 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
  • T Tao, Poincaré's legacies, pages from year two of a mathematical blog, II, Amer. Math. Soc., Providence, RI (2009)
  • C Viana, A note on the evolution of the Whitney sphere along mean curvature flow, preprint (2018)
  • Y L Xin, Translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015) 1995–2016