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An alternative is to simply use Riemann-Roch, which does not depend on the Uniformization Theorem. R-R says that $\ell(D)-\ell(K-D) = \mathrm{deg}(D)-g+1$, where $\ell(D)$ is the dimension of the s …

There is a classic paper by Phillip Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 77 …

Yes. The genus of $\Sigma$ is not really relevant. Here's an example: Let $f$ and $g$ be two meromorphic functions on $\Sigma$, where $g$ is nonconstant, and consider the sequence of maps $u_n: \Si …

I'm not sure what you mean by 'canonical'. For example, when $\Sigma$ has genus $1$, there are 4 distinct spin structures, one representing the trivial line bundle, whose space of holomorphic sectio …

Here is a bit more detail on the answer to the following question, which is my interpretation of what the OP is asking: Suppose that one knows the induced metric $g$ on a surface $S\subset\mathbb{R}^3 …

Actually, the rigidity result is local, and hence is even stronger: A very old result of Calabi implies that any connected (local) piece of a holomorphic curve in $\mathbb{CP}^n$ with constant Gaussi …

This is not really an answer, but rather a longish comment and a suggestion about how one might focus the question a bit better. First, when one asks for a 'canonical' isometric embedding into some E …

If a surface is foliated by geodesics (in the induced metric) that are also asymptotic lines, then these curves are also geodesics in the ambient manifold (just look at the definitions). Conversely, …