July/August 2024 The moment map on the space of symplectic 3d Monge-Ampère equations
Jan Gutt, Gianni Manno, Giovanni Moreno, Robert Śmiech
Adv. Differential Equations 29(7/8): 575-654 (July/August 2024). DOI: 10.57262/ade029-0708-575


For any $2{^\textrm{nd}}$ order scalar PDE $\mathcal{E}$ in one unknown function, we construct, by means of the characteristics of $\mathcal{E}$, a contact sub--bundle of the underlying contact manifold $J^1$, consisting of conic varieties, called the {contact cone structure} associated with $\mathcal{E}$. We then focus on symplectic Monge--Ampère equations in 3 independent variables, that are naturally parametrized, over ${\mathbb{C}}$, bythe projectivization of the 14--dimensional irreducible representation of the simple Lie group ${\mathsf{Sp}}(6,{\mathbb{C}})$. The associated moment map allows to define a rational map $\varpi$ from the space of symplectic 3D Monge-Ampère equations to the projectivization of the space of quadratic forms on a $6$--dimensional symplectic vector space. We study the relationship between the variety $\varpi({\mathcal{E}})=0$, herewith called the {co-characteristic variety} of ${\mathcal{E}}$, and the contact cone structure of a 3D Monge-Ampère equation ${\mathcal{E}}$, by obtaining a complete list of mutually non--equivalent quadratic forms on a $6$--dimensional symplectic space.


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Jan Gutt. Gianni Manno. Giovanni Moreno. Robert Śmiech. "The moment map on the space of symplectic 3d Monge-Ampère equations." Adv. Differential Equations 29 (7/8) 575 - 654, July/August 2024. https://doi.org/10.57262/ade029-0708-575


Published: July/August 2024
First available in Project Euclid: 29 January 2024

Digital Object Identifier: 10.57262/ade029-0708-575

Primary: 35A30 , 58A20 , 58J70

Rights: Copyright © 2024 Khayyam Publishing, Inc.


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Vol.29 • No. 7/8 • July/August 2024
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