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Holomorphically embedded discs with rapidly growing area. (English) Zbl 0929.32015

The note deals with closed complex submanifolds \(V\) of the open unit ball \(\mathbb B\subset\mathbb C^2\) which are biholomorphically equivalent to the open unit disc \(\Delta\subset\mathbb C\), \(V = f(\Delta),\) and \(f: \Delta\to \mathbb B\) is a proper holomorphic embedding. In particular, it is asked whether such a \(V\) can have infinite area. If one requires only that \(f\) is a proper holomorphic map and drops the requirement that \(f\) be an embedding, then the answer is positive. In fact, the area of \(V\cap r\mathbb B\) can grow arbitrarily rapidly as \(r\nearrow 1.\) To find an embedded disc is a much more delicate problem since the self-intersection points of \(f(\Delta)\) created in constructing \(f\) cannot be removed by small perturbations. Nevertheless, as the author shows, the following theorem holds: If \(\{r_n\}\) is a sequence of positive numbers that increases to 1 and if \(\{A_n\}\) is a sequence of real numbers, then there is a proper holomorphic embedding \(f:\Delta\to\mathbb B\) such that for all \(n\) the area of \(f(\Delta)\cap r_n\mathbb B\) is at least \(A_n.\)
It is also shown that if \(V\) is a closed submanifold of \(\mathbb C^2\) biholomorphically equivalent to the disc, \(f:\Delta\to\mathbb C^2,\) then the area of \(f(\Delta)\cap r_n\mathbb B\) can grow arbitrarily rapidly as \(r\nearrow+\infty.\)

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32H35 Proper holomorphic mappings, finiteness theorems
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[1] DOI: 10.1007/BF01444237 · Zbl 0859.32010 · doi:10.1007/BF01444237
[2] DOI: 10.1007/BF02559509 · Zbl 0853.58015 · doi:10.1007/BF02559509
[3] DOI: 10.2307/2374529 · Zbl 0678.32005 · doi:10.2307/2374529
[4] DOI: 10.2307/2372949 · Zbl 0104.05402 · doi:10.2307/2372949
[5] DOI: 10.1007/BF01461006 · Zbl 0847.32030 · doi:10.1007/BF01461006
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