Discs in hulls of real immersions to Stein manifolds
We obtain results on the existence of complex discs in plurisubharmonically convex hulls of Lagrangian and totally real immersions to Stein manifolds.
math.CV↗
arXiv subjects
Alexandre Sukhov
Publications and source records attributed to Alexandre Sukhov.
Pseudoholomorphic discs and symplectic structures in Hilbert space
We develop the theory of $J$-holomorphic discs in Hilbert spaces with almost complex structures. As an aplication, we prove a version of Gromov's symplectic non-squeezing theorem for Hilbert spaces. It can be applied to short-time symplectic flows of a wide class of Hamiltonian PDEs.
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Lagrangian inclusion with an open Whitney umbrella is rationally convex
It is shown that a Lagrangian inclusion of a real surface in $\mathbb C^2$ with a standard open Whitney umbrella and double transverse self-intersections is rationally convex.
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On the Hodge conjecture for $q$-complete manifolds
We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any $q$-complete complex manifold $X$ with $q<\dim X$. This holds in particular for the complement $X=\mathbb{C}\mathbb{P}^n\setminus A$ of any complex projective manifold defined by $q<n$ independent equations.
math.AG↗
Germs of singular Levi-flat hypersurfaces and holomorphic foliations
It is shown that the Levi foliation of a real analytic Levi-flat hypersurface extends to a $d$-web near a nondicritical singular point and admits a multiple-valued meromorphic first integral.
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Polynomially convex hulls of singular real manifolds
We obtain local and global results on polynomially convex hulls of Lagrangian and totally real submanifolds of $C^n$ with self-intersections and open Whitney umbrella points.
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Regularized maximum of strictly plurisubharmonic functions on an almost complex manifold
We prove that the maximum of two smooth strictly plurisubharmonic functions on an almost complex manifold can be uniformly approximated by smooth strictly plurisubharmonic functions.
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Gluing complex discs to Lagrangian manifolds by Gromov's method
The paper discusses some aspects of Gromov's theory.
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Local polynomial convexity of the unfolded Whitney umbrella in $\mathbb C^2$
The paper considers a class of Lagrangian surfaces in $\mathbb C^2$ with isolated singularities of the unfolded Whitney umbrella type. We prove that generically such a surface is locally polynomially convex near a singular point of this kind.
math.CV↗
Levi-flat filling of real two-spheres in symplectic manifolds (I)
Let (M,J,w) be a manifold with an almost complex structure J tamed by a symplectic form w. We suppose that M has complex dimension two, is Levi convex and has bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of M may be foliated by the boundaries of pseudoholomorphic discs.
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Stability of Stein structures on the Euclidean space
We give a characterization of the complex Euclidean space C^n, proving a global Newlander-Nirenberg's theorem on the representability of complex structures close to the standard complex structure on C^n.
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Proper J-holomorphic discs in Stein domains of dimension 2
We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2.
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Filling real hypersurfaces by pseudoholomorphic discs
We study pseudoholomorphic discs with boundaries attached to a real hypersurface in an almost complex manifold. We give sufficient conditions for filling a one sided neighborhood of the hypersurface by the discs.
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Filling hypersurfaces by discs in almost complex manifolds of dimension 2
We study pseudoholomorphic discs with boundaries attached to a real hypersurface in an almost complex manifold of dimension 2. We prove that if the hypersurface contains no discs, then they fill a one sided neighborhood of the hypersurface.
math.CV↗
Some aspects of analysis on almost complex manifolds with boundary
We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds and the elliptic regularity of some diffeomorphisms in almost complex manifolds with boundary.
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Extension of CR maps of positive codimension
We study the holomorphic extendability of smooth CR maps between real analytic strictly pseudoconvex hypersurfaces in complex affine spaces of different dimensions.
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Diffeomorphisms of Stein structures
We prove that a pseudoholomorphic diffeomorphism between two almost complex manifolds with boundaries satisfying some pseudoconvexity type condition cannot map a pseudoholomorphic disc in the boundary to a single point. This can be viewed as an almost complex analogue of a well-known theorem of J. E. Fornaess.
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Plurisubharmonic exhaustion functions and almost complex Stein structures
We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result for the study of convexity and hyperbolicity properties of these domains and the contact geometry of their boundaries.
math.CV↗