Vertex-minimal hyperbolic origami 2-torus

We show that there exists a geodesic triangulation $T$ of a hyperbolic genus 2 surface ${\mathrm{\Sigma }}_2$ with 10 vertices and an isometric polyhedral embedding $S:{\mathrm{\Sigma }}_2\hookrightarrow {\mathbb{H}}^3$ that sends the triangles in $T$ to geodesic triangles in ${\mathbb{H}}^3$. We call this type of embedding a hyperbolic origami 2-torus. Since 10 is the combinatorially minimum number of vertices required to triangulate a genus 2 surface, this paper settles the question of minimum number of vertices required to obtain a hyperbolic origami 2-torus.