Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with vertices at the singular points of the euclidean structure of $S$).
Question 1: can $S$ be realized (as a polyhedral surface) in an euclidean space? If yes, what is known about the set of such `euclidean polyhedral realizations'?
Question 2: same questions than above, but for $\mathcal P$ (i.e. an euclidean polyhedral decomposition of $S$ is fixed).
there is no assumption on the genus of $S$ (when $S$ has genus 0, an answer to the (first part of) Question 1 is given by Alexandrov theorem)
A similar question (but for Riemann surfaces) already appears in [B] (page 9). I guess that a lot has been done on this problem since. Any relevant reference would constitute an interesting answer.
Thanks for any help!
[B] Bers L., Riemann surfaces (1958)
[T] Troyanov M. -- Les surfaces a singularités coniques. Enseign. Math. 32 (1986), 79–94.