Survey of Oka theory.
Polynomially Convex Hulls with Piecewise Smooth Boundaries.
Mappings of quadric Cauchy-Riemann manifolds.
A reflection principle on strongly pseudoconvex domains with generic corners.
Extending proper holomorphic mappings of positive codimension.
Approximation by automorphisms on smooth submanifolds of C...
Analytic disks with boundaries in a maximal real submanifold of
Let be a two dimensional totally real submanifold of class in . A continuous map of the closed unit disk into that is holomorphic on the open disk and maps its boundary into is called an analytic disk with boundary in . Given an initial immersed analytic disk with boundary in , we describe the existence and behavior of analytic disks near with boundaries in small perturbations of in terms of the homology class of the closed curve in . We also prove a regularity theorem...
On the boundary regularity of proper mappings
Regularity of varieties in strictly pseudoconvex domains.
We prove a theorem on the boundary regularity of a purely p-dimensional complex subvariety of a relatively compact, strictly pseudoconvex domain in a Stein manifold. Some applications describing the structure of the polynomial hull of closed curves in C are also given.
Extending holomorphic mappings from subvarieties in Stein manifolds
Suppose that is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space to is a uniform limit of entire maps . We prove that a holomorphic map from a closed complex subvariety in a Stein manifold admits a holomorphic extension provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.
Holomorphic submersions from Stein manifolds
We establish the homotopy classification of holomorphic submersions from Stein manifolds to Complex manifolds satisfying an analytic property introduced in the paper. The result is a holomorphic analogue of the Gromov--Phillips theorem on smooth submersions.
On complete intersections
We construct closed complex submanifolds of which are differential but not holomorphic complete intersections. We also prove a homotopy principle concerning the removal of intersections with certain complex subvarieties of .
Oka manifolds: From Oka to Stein and back
Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations...
Discs in pseudoconvex domains.
Intersections of totally real and holomorphic disks.
It is shown that a holomorphically embedded open disk in C and a totally real embedded open disk which have a common smooth boundary have nontrivial intersection.
Localization of the Kobayashi Metric and the Boundary Continuity of Proper Holomorphic Mappings.
Approximation of biholomorphic mappings by automorphisms of Cn.
Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties
We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in ℂⁿ by Lempert and by Lárusson and Sigurdsson.
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