Isometric Embeddings of Locally Euclidean Metrics in $\mathbb R^3$ as Conical Surfaces

It is proved that if a domain with a locally Euclidean metric can be isometrically immersed in the Euclidean plane $\mathbb R^2$ with the standard metric, then it can be isometrically embedded in $\mathbb R^3$ as a conical surface whose projection on a sphere centered at the vertex of the cone is a self-avoiding planar graph with sufficiently smooth edges of specially selected lengths.