×

Lagrangian mean curvature flow of Whitney spheres. (English) Zbl 1432.53135

Let the orthogonal group \(\mathbb{O}(m)\) act on \(\mathbb{C}^m\) by \(A(x,y):=(Ax, Ay)\) for all \(A \in \mathbb{O}(m)\) and \(x, y \in \mathbb{R}^m\). An immersed Lagrangian submanifold in \(\mathbb{C}^m\) which is invariant under the group \(\mathbb{O}(m)\) is called equivariant. The Whitney sphere is an example. In this paper the authors study the Lagrangian mean curvature flow starting from an equivariant Lagrangian immersion of \(\mathbb{S}^m\) in \(\mathbb{C}^m\) with \(m>1\) such that the Ricci curvature satisfies a positivity condition. They show that the flow develops a type-II singularity which rescales to the product of a grim reaper with a flat Lagrangian subspace in \(\mathbb{C}^{m-1}\). The proof uses a detailed analysis of the evolution of the profile curves of the equivariant Lagrangian spheres.

MSC:

53E10 Flows related to mean curvature
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 10.4310/jdg/1214447218 · Zbl 0754.53006 · doi:10.4310/jdg/1214447218
[2] 10.1007/s10711-006-9082-z · Zbl 1098.35074 · doi:10.1007/s10711-006-9082-z
[3] 10.4310/jdg/1214446558 · Zbl 0731.53002 · doi:10.4310/jdg/1214446558
[4] ; Borrelli, C. R. Acad. Sci. Paris Sér. I Math., 321, 1485 (1995)
[5] 10.2748/tmj/1178225151 · Zbl 0877.53041 · doi:10.2748/tmj/1178225151
[6] 10.1007/s00209-009-0604-x · Zbl 1201.53075 · doi:10.1007/s00209-009-0604-x
[7] 10.1007/s00209-007-0126-3 · Zbl 1144.53084 · doi:10.1007/s00209-007-0126-3
[8] 10.4310/jdg/1214456010 · Zbl 0827.53006 · doi:10.4310/jdg/1214456010
[9] 10.4310/jdg/1271271795 · Zbl 1206.53071 · doi:10.4310/jdg/1271271795
[10] 10.1007/s00526-015-0886-2 · Zbl 1336.53081 · doi:10.1007/s00526-015-0886-2
[11] 10.1007/s00222-007-0036-3 · Zbl 1119.53052 · doi:10.1007/s00222-007-0036-3
[12] 10.1090/S0002-9947-2013-05649-8 · Zbl 1282.53057 · doi:10.1090/S0002-9947-2013-05649-8
[13] 10.2969/jmsj/05010203 · Zbl 0906.53037 · doi:10.2969/jmsj/05010203
[14] 10.1007/BF01189952 · Zbl 0921.53025 · doi:10.1007/BF01189952
[15] 10.1007/s00526-018-1458-z · Zbl 1406.53076 · doi:10.1007/s00526-018-1458-z
[16] 10.1007/s00526-015-0853-y · Zbl 1325.53091 · doi:10.1007/s00526-015-0853-y
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.