2134/27771
B. Doubrov
B.
Doubrov
Evgeny Ferapontov
Evgeny
Ferapontov
B. Kruglikov
B.
Kruglikov
Vladimir Novikov
Vladimir
Novikov
Integrable systems in four dimensions associated with six-folds in Gr(4, 6)
Loughborough University
2017
Submanifold of the Grassmannian
Dispersionless integrable system
Hydrodynamic reduction
Self-dual conformal structure
Monge-Ampere system
Dispersionless lax pair
Linear degeneracy
Mathematical Sciences not elsewhere classified
2017-12-13 14:34:52
Journal contribution
https://repository.lboro.ac.uk/articles/journal_contribution/Integrable_systems_in_four_dimensions_associated_with_six-folds_in_Gr_4_6_/9384998
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.