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Symmetry, Integrability and Geometry: Methods and Applications, 2015, Volume 11, 081, 32 pp.
DOI: https://doi.org/10.3842/SIGMA.2015.081
(Mi sigma1062)
 

This article is cited in 2 scientific papers (total in 2 papers)

Monge–Ampère Systems with Lagrangian Pairs

Goo Ishikawaa, Yoshinori Machidab

a Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
b Numazu College of Technology, 3600 Ooka, Numazu-shi, Shizuoka, 410-8501, Japan
Full-text PDF (499 kB) Citations (2)
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Abstract: The classes of Monge–Ampère systems, decomposable and bi-decomposable Monge–Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss–Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables $\geq 3$. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge–Ampère systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge–Ampère systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3).
Keywords: Hessian Monge–Ampère equation; non-degenerate three form; bi-Legendrian fibration; Lagrangian contact structure; geometric structure; simple graded Lie algebra.
Received: April 10, 2015; in final form October 5, 2015; Published online October 10, 2015
Bibliographic databases:
Document Type: Article
MSC: 58K20; 53A15; 53C42
Language: English
Citation: Goo Ishikawa, Yoshinori Machida, “Monge–Ampère Systems with Lagrangian Pairs”, SIGMA, 11 (2015), 081, 32 pp.
Citation in format AMSBIB
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\paper Monge--Amp\`ere Systems with Lagrangian Pairs
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\vol 11
\papernumber 081
\totalpages 32
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Symmetry, Integrability and Geometry: Methods and Applications
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