Bulletin of the Belgian Mathematical Society - Simon Stevin

Lagrangian submanifolds in para-complex Euclidean space

Henri Anciaux and Maikel Antonio Samuays

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Abstract

We address the study of some curvature equations for distinguished submanifolds in para-Kähler geometry. We first observe that a para-complex submanifold of a para-Kähler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space $\mathbb{D}^n$ and discuss a number of examples, such as graphs and normal bundles. We also characterize those Lagrangian surfaces of $\mathbb{D}^2$ which are minimal and have indefinite metric. Finally we describe those Lagrangian self-similar solutions of the Mean Curvature Flow (with respect to the neutral metric of $\mathbb{D}^n$) which are $SO(n)$-equivariant.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 23, Number 3 (2016), 421-437.

Dates
First available in Project Euclid: 6 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1473186515

Digital Object Identifier
doi:10.36045/bbms/1473186515

Mathematical Reviews number (MathSciNet)
MR3545462

Zentralblatt MATH identifier
1351.53095

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53D12: Lagrangian submanifolds; Maslov index

Keywords
para-Kähler geometry Lagrangian submanifolds Minimal submanifolds Self-similar solutions to the Mean Curvature Flow

Citation

Anciaux, Henri; Samuays, Maikel Antonio. Lagrangian submanifolds in para-complex Euclidean space. Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 3, 421--437. doi:10.36045/bbms/1473186515. https://projecteuclid.org/euclid.bbms/1473186515


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