Papers by Sergey Ivashkovich
Our goal here is to prove that each point in an almost-complex surface has a basis of complete hy... more Our goal here is to prove that each point in an almost-complex surface has a basis of complete hyperbolic neighborhoods. The problem is local, and therefore we can consider the case when our surface is R 4 with an arbitrary almost-complex structure J. Throughout this paper, almost-complex structures are assumed to be of class C 1,α for some 0 < α < 1. Let C be some non-singular J-complex curve passing through the origin.
Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls
Inventiones Mathematicae, 1999
Mathematische Zeitschrift, 2011
We prove the existence of primitive curves and positivity of intersections of J-complex curves fo... more 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).
We prove the existence of primitive curves and positivity of intersections of J-complex curves fo... more 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 C 1,Lip . We also give the optimal regularity of curves in Lipschitz structures. It occurs to be 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).
After the "inner" case considered in Theorem 1, we prove in § 5 the compactness theorem for curve... more After the "inner" case considered in Theorem 1, we prove in § 5 the compactness theorem for curves with boundary on totally real submanifolds. For this "boundary" case we give appropriate generalizations of all "inner" constructions and estimates.
Continuity Principle And Extension Properties Of Meromorphic Mappings With Values In Non Kähler Manifolds
Upper semi-continuity of the Kobayashi-Royden pseudo-norm, a counterexample for Hölderian almost complex structures
Arkiv for Matematik, 2005
ABSTRACT If $X$ is an almost complex manifold, with an almost complex structure $J$ of class $\CC... more ABSTRACT If $X$ is an almost complex manifold, with an almost complex structure $J$ of class $\CC^\alpha$, for some $\alpha &gt;0$, for every point $p\in X$ and every tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through $p$ with this prescribed tangent vector. This existence result goes back to Nijenhuis-Woolf. All the $J$ holomorphic curves are of class $\CC^{1,\alpha}$ in this case. Then, exactly as for complex manifolds one can define the Royden-Kobayashi pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is an upper semi-continuous function on the tangent bundle. For complex manifolds it is the crucial point in Royden&#39;s proof of the equivalence of the two standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for structures that are smooth enough. In [I-R], it is shown that $\CC^{1,\alpha}$ regularity of $J$ is enough. Here we show the following: Theorem. There exists an almost complex structure $J$ of class $\CC^{1\over 2}$ on the unit bidisc $\D^2\subset \C^2$, such that the Royden-Kobayashi seudo-norm is not an upper semi-continuous function on the tangent bundle.
We establish a Fatou-type Theorem for J-holomorphic mappings that are bounded in an appropriate s... more We establish a Fatou-type Theorem for J-holomorphic mappings that are bounded in an appropriate sense and we prove the Blaschke condition for their zero sets. We also prove a Privalov-type uniqueness Theorem for pairs of J-holomorphic mappings.
We establish the Schwarz Reflection Principle for J-complex discs attached to a real analytic J-t... more We establish the Schwarz Reflection Principle for J-complex discs attached to a real analytic J-totally real submanifold of an almost complex manifold with real analytic J. We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem in C k,α -classes 1 .
We establish the Schwarz Reflection Principle for J-complex discs attached to a real analytic J-t... more We establish the Schwarz Reflection Principle for J-complex discs attached to a real analytic J-totally real submanifold of an almost complex manifold with real analytic J. As a second result a boundary uniqueness theorem for J-complex discs with Lipschitz-continuous J is obtained. We also prove the precise regularity of J-complex discs attached to a J-totally real submanifold 1 .
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Papers by Sergey Ivashkovich