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

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