### Survey of Oka theory.

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Let $M$ be a two dimensional totally real submanifold of class ${C}^{2}$ in ${\mathbf{C}}^{2}$. A continuous map $F:\stackrel{\u203e}{\Delta}\to {\mathbf{C}}^{2}$ of the closed unit disk $\stackrel{\u203e}{\Delta}\subset \mathbf{C}$ into ${\mathbf{C}}^{2}$ that is holomorphic on the open disk $\Delta $ and maps its boundary $b\Delta $ into $M$ is called an analytic disk with boundary in $M$. Given an initial immersed analytic disk ${F}^{0}$ with boundary in $M$, we describe the existence and behavior of analytic disks near ${F}^{0}$ with boundaries in small perturbations of $M$ in terms of the homology class of the closed curve ${F}^{0}\left(b\Delta \right)$ in $M$. We also prove a regularity theorem...

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.

Suppose that $Y$ is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space ${\u2102}^{n}$ to $Y$ is a uniform limit of entire maps ${\u2102}^{n}\to Y$. We prove that a holomorphic map ${X}_{0}\to Y$ from a closed complex subvariety ${X}_{0}$ in a Stein manifold $X$ admits a holomorphic extension $X\to Y$ provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.

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.

We construct closed complex submanifolds of ${\u2102}^{n}$ which are differential but not holomorphic complete intersections. We also prove a homotopy principle concerning the removal of intersections with certain complex subvarieties of ${\u2102}^{n}$.

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...

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.

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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