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I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow. I've just come across a Twitter thread by Laurent Fargues explaining a work ...

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...