Suppose $\varphi:\mathbb CP^1\to \mathbb CP^1$ is a homeomorphism holomorphic on a connected open subset $U\subset \mathbb CP^1$ with $\mathbb CP^1\setminus U$ of zero measure.

Is it true that $\varphi$ is holomorphic on the whole $\mathbb CP^1$ (so it is a projective transformation)?

If no, what kind of assumptions of $U$ would suffice? (for example $\mathbb CP^1\setminus U$ has Hausdorff dimension $\le 1$?)

  • 2
    $\begingroup$ Answers here should be relevant. $\endgroup$
    – Wojowu
    May 24 at 22:09

Following the links of Wojowu, the answer to this question is negative for the case of self-homeomorphisms of $\mathbb C^1$, here it is:

Functions holomorphic on a region minus a Cantor set

So by extending the self-homeo to $\mathbb CP^1$ the answer is negative for $\mathbb CP^1$ as well. To have a positve answer, one has to require indeed that the Hausdorff dimension of $\mathbb CP^1\setminus U$ is less than $1$. (the case of $\dim=1$ seems to be still open)


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