Embedding of flat surfaces

Question

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).

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Remarks:

• 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!

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[B] Bers L., Riemann surfaces (1958)

[T] Troyanov M. -- Les surfaces a singularités coniques. Enseign. Math. 32 (1986), 79–94.

Answer 1

It may be that this theorem of Burago & Zalgaller (partially) answers your question?

Theorem (Burago-Zalgaller 1.7). Every polyhedron $M$ admits an isometric piecewise-linear $C^0$ immersion into $\mathbb{R}^3$. If $M$ is orientable or has a nonempty boundary, then $M$ admits an isometric piecewise-linear $C^0$ embedding into $\mathbb{R}^3$.

This is from:

• Yu. D. Burago and V. A. Zalgaller. "Isometric piecewise linear immersions of two-dimensional manifolds with polyhedral metrics into $\mathbb{R}^3$. St. Petersburg Math. J., 7(3):369--385, 1996. Translated by S. G. Ivanov. English translation: Scanned PDF (15MB).

Here is their definition of a "polyhedron":

By a two-dimensional manifold with polyhedral metric (in brief, a polyhedron), we mean a metric space endowed with the structure of a connected compact two-dimensional manifold (possibly with boundary) every point $x$ of which has a neighborhood isometric to the vertex of a cone. ... The metric is locally flat everywhere except for a finite collection of points; these points are the `true' vertices."

A nice phrase they use in Lemma 2.2 to describe the mapping is that each triangle becomes a pleated surface in $\mathbb{R}^3$.