Here is a polygonal disk + gluing scheme model of a surface we are attempting to construct. We want the regions of the surface bounded by the two vertical, dotted lines $\alpha,\beta$ to have zero curvature throughout, while the rest of the surface should be "smoothed out" with some kind of metric which gives it negative curvature and thus allows the entire shape to satisfy the requirement that a two-holed torus have negative total curvature.

Assuming that we can adjust the circled (and, separately, squared) angles so that they add up to $2\pi$ and the relevant identifications (side $a$ with side $a$, $b$ with $b$) can be made smoothly while maintaining the zero curvature of the inner region, the question is the following. Will the smoothing out of edges $1$ through $4$ necessarily need to be continued into the region inside of which we want to have zero curvature, thus giving it non-zero curvature instead? In other words, we *have* to smooth out the triangled vertices to *some* extent because their corresponding angles add up to more than $2\pi$. But, thinking geometrically (we have a limited knowledge of the underlying Riemannian geometry here), it seems that we would need to smooth out *neighborhoods* of these vertices and of the edges $1$ through $4$. Will we be able to stop before reaching the region inside of $\alpha,\beta$?

Thank you. Any references that might be able to help us are welcome. We're pretty new to this but if the answer to these questions is that we can maintain zero curvature, we've made progress on our problem!