Differential invariants for cubic integrals of geodesic flows on surfaces

We construct differential invariants that vanish if and only if the geodesic flow of a two-dimensional metric admits an integral of third degree in momenta with a given Birkhoff–Kolokoltsov 3-codifferential.     

Limit holonomy and extension properties of Sobolev and Yang-Mills bundles

We consider the local behavior of Sobolev connections in a neighborhood of a singularity of codimension 2 and determine sufficient conditions for existence and local constancy of the limit holonomy of such connection. We prove that every Sobolev connection on an mdimensional manifold with locally L^m/2-integrable curvature and trivial limit holonomy extends through singularity of codimension 2. Additionally, if the connection satisfies the Yang-Mills-Higgs equation, the extension also satisfies the equation.     

Cohomological barriers to the extension of holomorphic line bundles

For holomorphic line bundles with L² -bounded curvature we prove an theorem on extension across an analytic subset and construct a counterexample to extension across a smooth noncomplex submanifold of codimension 2.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 4–8.     

Local properties of J-complex curves in Lipschitz-continuous structures

We prove the existence of primitive curves and positivity of intersections of J-complex curves for Lipschitz-continuous almost complex structures. These results are deduced from the Comparison Theorem for J-holomorphic maps in Lipschitz structures, previously known for J of class ${\mathcal{C}^{1, Lip}}$ . We also give the optimal regularity of curves in Lipschitz structures. It occurs to be ${\mathcal{C}^{1,LnLip}}$ , i.e. the first derivatives of a J-complex curve for Lipschitz J are Log-Lipschitz-continuous. A simple example that nothing better can be achieved is given. Further we prove the Genus Formula for J-complex curves and determine their principal Puiseux exponents (all this for Lipschitz-continuous J-s).     

The Darboux-Type Theorems for ∂-Symplectic and ∂-Contact Structures

A -symplectic structure on a complex manifold M of complex dimension2n is given by a smooth -closed (2, 0)-form such that ⁿ is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally can be represented as _i=1 ⁿ f _i f _{n + i} for appropriate smooth complex valuedfunctions f ₁, ..., f _2n . We also present a contact version of this theorem.     

Two-dimensional superintegrable metrics with one linear and one cubic integral

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta.