# Self homeomorphism of $\mathbb CP^1$ holomorphic a.e

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 $$\le 1$$?)

• Answers here should be relevant. 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:
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)