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Boundary regularity for holomorphic maps from the disc to the ball. (English) Zbl 0611.30026
Let U be the open unit disc in $${\mathbb{C}}$$. In the paper the authors prove that any holomorphic map $$f: U\to {\mathbb{C}}$$ whose global cluster set has finite length and whose Dirichlet integral $$\iint _{U}| f'| ^ 2$$ is finite extends continuously to Ū. They apply this to study the boundary regularity of proper holomorphic maps from U to the ball in $${\mathbb{C}}^ n$$. The authors’ conjecture that in the above result it is not necessary to assume that the Dirichlet integral is finite has since been proved by H.Alexander [Polynomial hulls and linear measure, to appear] and C. Pommerenke [Mich. Math. J. 34, 93-97 (1987)].

##### MSC:
 30D40 Cluster sets, prime ends, boundary behavior
##### Keywords:
global cluster set; Dirichlet integral
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