Boris Kruglikov

Geometry of differential equations

We review geometric and algebraic methods of investigations of systems of partial differential equations. Classical and modern approaches are reported.

Invariant characterization of Liouville metrics and polynomial integrals

Dette er forfatternes aksepterte versjon. This is the author’s final accepted manuscript.

Invariants of pseudogroup actions: Homological methods and Finiteness theorem

We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of l-variants and l-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: we introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie–Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations.

The gap phenomenon in parabolic geometries

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostants version of the Bott-Borel-Weil theorem to show that this bound is in fact sharp in almost all complex and split-real cases by exhibiting (abstract) models. We explicitly compute all submaximal symmetry dimensions when G is any complex or split-real simple Lie group.

Point classification of 2nd order ODEs:Tresse classification revisited and beyond

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the second order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.

Mayer brackets and solvability of PDEs -- II

A solvability of a system of partial differential equations with two variables is investigated. The smooth solvability conditions are formulated in terms of (generalized) Mayer brackets.  2002 Elsevier Science B.V. All rights reserved.

Mayer brackets and solvability of PDEs—I

Abstract A solvability of a system of partial differential equations with two variables is investigated. The smooth solvability conditions are formulated in terms of (generalized) Mayer brackets.

Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations

We study integrable non-degenerate Monge–Ampère equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining those equations. This knowledge is used to deform these heavenly type equations into new integrable PDEs of the second-order with large symmetry pseudogroups. We classify the symmetric deformations obtained in this way and discuss self-dual hyper-Hermitian geometry of their solutions, thus encoding integrability via the twistor theory.

On the Einstein-Weyl and conformal self-duality equations

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as “master dispersionless systems” in four and three dimensions, respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note, we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar partial differential equations for a general anti-self-dual conformal structure in neutral signature.

Pseudoholomorphic mappings and Kobayashi hyperbolicity

Abstract We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familiar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold. We give special consideration to almost complex structures tamed by some symplectic form. The notions and pseudoholomorphic curves involved are illustrated in some examples.