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Ali Taghavi
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Holomorphic Gauss normal map

Let $S$ be a Riemann surface smoothly embedded in $\mathbb{R}^3$.

Is there necessarily a smooth embedding of $S$ in $\mathbb{R}^3$ such that the Gauss normal map $n:S \to S^2$ would be a holomorphic map?Under what conditions on $S$ is the answer affirmative?

The question asks:can we change smoothly the position of a Riemann surface to have a holomorphic Gauss map?

Note: The sphere is considered as a Riemann surface with its standard structure

Ali Taghavi
  • 356
  • 8
  • 31
  • 123