Holomorphic 1-forms On The Riemann Sphere

Differential Geometry
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Tsakanikas Nickos
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Holomorphic 1-forms On The Riemann Sphere

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Post by Tsakanikas Nickos »

Show that a holomorphic 1-form \( \displaystyle \omega = f \mathrm{d}z \) on \( \mathbb{C} \) extends to a holomorphic 1-form on \( \mathbb{\hat{C}} \) if and only if \( \displaystyle \lim_{z \to \infty} z^{2}f(z) \) is finite. Conclude that there are no non-zero holomorphic 1-forms on \( \mathbb{\hat{C}} \).
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