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Suppose $M$ and $N$ are two Stein manifolds of dimension at least $3$ with compact subsets $U$ and $V$ such that $M\setminus U$ is biholomorphic to $N \setminus V$. It it true that $M$ is biholomorphic to $N$? It this is not true, what is the simplest example? And if this is true, what would be the reference for such a statement? |
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I believe the answer is positive in dimension at least two. Stein manifolds admit proper embeddings on vector spaces. An isomorphism from $M$ to $N$ can be represented by a collection of holomorphic functions from $M$ to $\mathbb C$. Each one of these extends to the whole $M$ according to Hartogs. Thus the isomorphism at infinity extends to a holomorphic map. Arguing in the same way with the inverse of the isomorphism at infinity yields an affirmative answer. |
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