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Title: Integrable systems in 4D associated with sixfolds in Gr(4; 6)
Authors: Doubrov, B.
Ferapontov, E.V.
Kruglikov, B.
Novikov, V.S.
Keywords: Submanifold of the Grassmannian
Dispersionless integrable system
Hydrodynamic reduction
Self-dual conformal structure
Monge-Ampere system
Dispersionless lax pair
Linear degeneracy
Issue Date: 2018
Publisher: Oxford University Press
Citation: DOUBROV, B. ...et al., 2018. Integrable systems in 4D associated with sixfolds in Gr(4; 6). International Mathematics Research Notices, In Press.
Abstract: Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V . A submanifold X ⇢ Gr(d, n) gives rise to a di↵erential system ⌃(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a sixfold in Gr(4, 6). The corresponding system ⌃(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems ⌃(X). These naturally fall into two subclasses. • Systems of Monge-Ampere type. The corresponding sixfolds X are codimension 2 linear sections of the Pl¨ucker embedding Gr(4, 6) ,! P14. • General linearly degenerate systems. The corresponding sixfolds X are the images of quadratic maps P6 99K Gr(4, 6) given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system ⌃(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.
Description: This paper is in closed access until 12 months after publication.
Sponsor: The research of EVF was partially supported by the EPSRC grant EP/N031369/1.
Version: Accepted for publication
URI: https://dspace.lboro.ac.uk/2134/27771
Publisher Link: https://academic.oup.com/imrn
ISSN: 1073-7928
Appears in Collections:Closed Access (Maths)

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