arXiv subjects
Publications and source records attributed to Alexandre Sukhov.
We prove an analogue of the Brody lemma in the framework of Riemannian manifolds. We also present new examples of Riemannian manifolds that are hyperbolic in the sense of Kobayashi.
We study the boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds. As an application, we prove a Fatou type theorem on the existence, almost everywhere, of non tangential limits for bounded conformal harmonic immersed discs. We also prove a Picard theorem for conformal harmonic discs and give some examples of Kobayashi hyperbolic Riemannian manifolds.
We introduce an analog of the Kobayashi-Royden metric on a Riemannian manifold and study its basic properties.
We prove a Fatou type theorem for bounded functions with d_J -bar differential of a controled growth on smoothly bounded domains in an almost complex manifold.
We prove a version of the Fatou theorem for bounded functions with bounded d_J-bar diferential on wedge-type domains in an almost complex manifold.
We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$-boundaries.
We discuss local polynomial convexity of real analytic Levi-flat hypersurfaces in $\mathbb C^n$, $n>1$, near singular points.
We study proper holomorphic mappings between strictly pseudoconvex domains with low boundary regularity.
We prove a Wong-Rosay type theorem for a domain with a piecewise smooth generic strictly pseudoconvex boundary point.
We prove analogs of the Chirka - Lindelof and Fatou theorems for bounded functions with bounded d-bar on a strictly pseudoconvex domain in an almost complex manifold
We study boundary properties of plurisubharmonic functions near real submanifolds of almost complex manifolds.
We prove that a totally real manifold (of maximal dimension) is a boundary uniqueness set for a psh function on an almost complex manifold.
It is proved that dicritical singularities of real analytic Levi-flat sets coincide with the set of Segre degenerate points.
We establish an effective criterion for a dicritical singularity of a real analytic Levi-flat hypersurface. The criterion is stated in terms of the Segre varieties. As an application, we obtain a structure theorem for some class of currents in the nondicritical case.
This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.
In this paper, we give two direct applications of the theory of singular connections developped by Harvey-Lawson [10]. The first one is a version of Lelong-Poincaré formula for vector bundle over an almost complex manifold. The second is a convergence theorem for divisors associated to symplectic submanifolds constructed by Auroux in [2]. The case of hypersurfaces was done by Donaldsson in [4].
We show that a Lagrangian inclusion in $\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\mathbb RP_2$, admits a pair of smooth complex-valued functions $f_1$, $f_2$ with the property that any continuous complex valued function on $S$ is a uniform limit of a sequence of $R_j(f_1,f_2)$, where $R_j(z_1,z_2)$ are rational functions in $\mathbb C^2$.
We prove a generalization of Gromov's symplectic non-squeezing theorem for the case of Hilbert spaces. Our approach is based on filling almost complex Hilbert spaces by complex discs partially extending Gromov's results on existence of $J$-complex curves. We apply our result to the flow of the discrete nonlinear Schrödinger equation.