Real-analytic hypersurfaces in complex manifolds

At the beginning of this century Hartogs $\lbrack15\rbrack$ directed attention to an interesting property of holomorphic functions of several complex variables, namely, their continuability. For example, if a function is holomorphic on the boundary of a ball in $\mathbf C^n$ ($n\geq2$), then it can be continued to a function holomorphic in the whole ball.
Holomorphic maps from $\mathbf C^n$ into $\mathbf C^n$ have greater inflexibility than functions, and this is the basis of several interesting facts. A typical situation that illustrates the continuation property of maps is a theorem of Poincaré $\lbrack1\rbrack$. If in a neighbourhood of a point of the sphere $S_{2n-1}\subset\mathbf C^n$ there is defined biholomorphic map sending the sphere into itself, then it can be continued holomorphically to the whole sphere, and moreover it turns out to be a linear-fractional transformation. After Poincaré this theorem was stated by other authors ($\lbrack11\rbrack$, $\lbrack44\rbrack$, and $ \lbrack12\rbrack$).
Now let $M$ be a strictly pseudoconvex compact real-analytic hypersurface in $\mathbf C^n$, and $D$ the domain bounded by it (a surface is called strictly pseudoconvex if it can be transformed into a strictly convex surface by a biholomorphic change of variables close to any point of it). By Hartog's theorem every automorphism of $M$ can be continued holomorphically to $D$. Can we expect every automorphism of a surface to be continued holomorphically to some domain lying on the outside of $M$? For spheres this is not true; for any point lying outside a sphere we can find an automorphism of the sphere for which this point is singular. But it turns out that a sphere is in a certain sense a unique exception; if $M$ is not (locally) equivalent to a sphere, then the answer to our question is positive, that is, all automorphisms of $M$ can be continued holomorphically to a common neighbourhood of $M$ (see Corollary 3 to the theorem in § 8). We note that the part of this neighbourhood lying outside $M$ does not belong to the hull of holomorphy of $M$, and so functions holomorphic in $M$ cannot, in general, be continued to this part of the neighbourhood.
These examples are only fascinating fragments of the trend in modern complex analysis discussed in this paper. We discuss the basic stages in the development of this trend, concentrating attention on recent results. Our subject matter is hypersurfaces in $\mathbf C^n$ and in other complex manifolds that are non-degenerate in the sense of Levi, the forms of representing these surfaces, their automorphism groups, the geometry of Chern–Moser chains, the continuation of holomorphic maps, questions of classifications, and others.