## Geometry & Topology

### Lagrangian mean curvature flow of Whitney spheres

#### Abstract

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

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

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

https://projecteuclid.org/euclid.gt/1555466435

Digital Object Identifier
doi:10.2140/gt.2019.23.1057

Mathematical Reviews number (MathSciNet)
MR3939057

Zentralblatt MATH identifier
07056058

#### Citation

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. https://projecteuclid.org/euclid.gt/1555466435

#### References

• 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 http://nbn-resolving.de/urn:nbn:de:gbv:089-5378074038 {\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