Some aspects of holomorphic mappings: A survey

Abstract

Abstract. This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.

Embedding bordered Riemann surfaces in strongly pseudoconvex domains

“…(Recall that a CR map is one that satisfies the tangential Cauchy-Riemann equations, and in the case of a strongly pseudoconvex source hypersurface such maps extend locally holomorphically to the pseudoconvex side.) The question which (strongly) pseudoconvex hypersurfaces admit local CR embeddings into various model hypersurfaces has attracted considerable attention; see the surveys by Baouendi, Ebenfelt and Rothschild [4], Pinchuk, Shafikov, and Sukhov [35], and (for maps to balls) by D'Angelo [11]. On the other hand, there are proper holomorphic maps and embeddings of strongly pseudoconvex domains to balls and other model domains which extend continuously to the closure of the domain; see the papers by Løw [33], Globevnik [29], Hakim [30], and Dor [13], among others.…”

Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

“…Euclidean spaces has been studied extensively; see the recent survey by Pinchuk et al [93]. It is generally believed, and has been proved under a variety of additional conditions, that proper holomorphic maps between relatively compact smoothly bounded domains of the same dimension always extend smoothly up to the boundary.…”

“…I have not included any topics from Cauchy-Riemann geometry since it would be impossible to properly discuss this major subject in the present survey of limited size and with a rather different focus. The reader may wish to consult the recent survey by Pinchuk et al [93], the monograph by Baouendi et al [18] from 1999, and my survey [48] from 1993. For a new direction in this field, see the papers by Bracci and Gaussier [27,26].…”

Abstract boundaries and continuous extension of biholomorphisms

“…Extensions of biholomorphisms between domains is a classical subject which has been attacked and studied by several authors with different techniques. It is virtually impossible to name all authors which contributed to the subject, and we limit ourselves in citing the surveys papers [PSS,Fo,Fo1] and references therein. In some cases, the problem of extension is dealt with techniques of CR geometry and asymptotic expansions of Bergman kernels or other invariant metrics, like the amazing result of Ch.…”