Proper holomorphic mappings.

Holomorphic Mappings between Domains in ℂ2

“…Arguments similar to those in [13,21] show that ψ p (z ′ ) is a negative, continuous plurisubharmonic function on D ′ . Furthermore there is an open neighbourhood U ′ of p ′ small enough so that U ′ ∩ M ′ is smooth real analytic such that…”

“…The theorem above shows that it is possible to study the boundary behaviour of f under purely local hypotheses. Motivated in part by [2,13,21], which deal with similar local theorems under convexity assumptions on the boundaries either of a geometric or a function theoretic nature, attempts to arrive at such a local statement were made in [23,25], both of which were proved under additional hypotheses only on the mappings involved. It should be noted that local extension theorems for proper holomorphic mappings across pseudoconvex real analytic boundaries played a particularly useful role in the global theorem of [9].…”

Some aspects of holomorphic mappings: A survey

“…Various results concerning continuous extension of holomorphic mappings are also obtained in [24,25,60,73,84].…”

Uniformization of strictly pseudoconvex domains. I

“…Then the germ of the map π −1 extends to a biholomorphic map g from U ∩ D to an open set in B. A standard argument using the Hopf lemma and the asymptotics of the Poincaré metric on B (see, e. g., [18], [9] or [25]) shows that g extends to ∂D ∩ U as a Hölder continuous map sending ∂D ∩ U to the unit sphere. Note that by the boundary uniqueness theorem the extension to the boundary is not constant.…”