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Let $H^2$ be the polydisk given by the set of points $(z,w)$ in $\mathbb C^2$ with by $\Im(z) > 0$ and $\Im(w) > 0$.

Consider the surface $S$ defined by $z^2 + w^2 = -1$ within $H^2$. On this surface $z$ is a good coordinate (although points on the line segment $\ell$ given by $z = i y$ with real $y > 1$ are not part of $S$ since for these points $w$ is no longer in $H$).

Consider now a holomorphic function $f(z)$ on $S$. Does there exist a holomorphic function $\hat f(z,w)$ on $H^2$ such that $f(z) = \hat f(z,w)|_{S}$ ?

My formulation hopefully makes it clear that the function $\hat f(z,w)$ is not necessarily an analytic continuation. Notice further that a constant extension, so defining $\hat f(z,w) := f(z)$, will not generally work; an example would be $f(z) = \sqrt{z - i}$ or any other function with a cut along $\ell$.

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    $\begingroup$ The domain $H^2$ is a Stein manifold (since it has a Stein exhaustion), and $S$ is a closed submanifold in $H^2$, so by general theorems about the vanishing of higher coherent analytic sheaf cohomology every analytic function on $S$ extends to one on $H^2$. More concretely, for the sheaf $O_S$ of analytic functions on $S$ and $O_{H^2}$ likewise for $H^2$ we have a short exact sequence of sheaves $0 \rightarrow O_{H^2} \rightarrow O_{H^2} \rightarrow O_S \rightarrow 0$ where the first map is multiplication by $z^2 + w^2 + 1$. Thus, the vanishing of ${\rm{H}}^1(H^2, O_{H^2})$ does the job. $\endgroup$
    – nfdc23
    Commented Nov 13, 2016 at 21:21
  • $\begingroup$ @nfdc23 Thank you very much, this sounds very helpful although somewhat above my current pay grade - but now I at least where to look. I'd be happy to accept your comment as an answer. Do you think the same is true for the higher-dimensional case, i.e. for the submanifold given by z_1^2 + ... + z_n^2 = - 1 in H^n? $\endgroup$
    – fanfare
    Commented Nov 13, 2016 at 21:43
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    $\begingroup$ Yes, the theory of Stein spaces (and that argument) works in all dimensions. The book "Theory of Stein Spaces" by Grauert and Remmert is extremely well-written, say building on some basic experience with coherent analytic sheaves from their other book "Coherent Analytic Sheaves" (don't need a lot from there, say just up through Oka's theorem on coherence of the sheaf of holomorphic functions on $\mathbf{C}^n$). It is well worth the effort for understanding the modern way of dealing with classical construction problems in holomorphic function theory (such as topological obstructions, etc.). $\endgroup$
    – nfdc23
    Commented Nov 13, 2016 at 23:06

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