The side-lengths of the simplicial complex are given. Alternatively, only the angles of the triangles are given, and the triangulated 2-sphere is known up to change of scale. By a canonical homeomorphism, I mean a homeomorphism that is specified, up to composition with an element of the orthogonal group $O(3)$.
Using the euclidean path-metric on the simplicial complex, one obtains a flat 2-sphere, except for a finite number of cone points.
I have been told that some version of the Hamilton-Perelman Ricci flow will take the flat 2-sphere with cone points to the round 2-sphere. (Is this known/published?) I know next to nothing about the Ricci flow. However I have the feeling the resulting homeo will not be specified sufficiently for it to be useful for the sort of applications I have in mind.
One approach to my problem (which I have been unable to carry out) might be the following. Let ${c_1,...,c_k}$ be the cone points on the flat 2-sphere, and let $C_i$ consist of points whose nearest cone point is $c_i$, like the Delaunay/Voronoi/Dirichlet construction. Then $C_i$ is a polygon, with a finite number of straight boundary edges. The vertices on the boundary of $C_i$ are points that are equidistant from three or more cone points. We triangulate the flat 2-sphere as follows. For each $i$, we triangulate $C_i$ by joining $c_i$ to each of the boundary vertices of $C_i$.
It would be great if one could, for example, reassign spherical angles or spherical lengths to the simplicial complex just described, so as to obtain curvature +1. I'm hoping that the spherical angles might be functions of the angles occuring in the triangulation just described of the flat 2-sphere with cone points.
A special case might be easier to think about. Suppose the triangulated 2-sphere can be expressed as the union of cubes in a cubical complex. Then the only possible cone angles are $3\pi/2$ and $3\pi$.