On triangle meshes with valence 6 dominant vertices

Jean-Marie Morvan
2018 unpublished
We study triangulations T defined on a closed disc X satisfying the following condition : In the interior of X, the valence of all vertices of T except one of them (the irregular vertex) is 6. By using a flat singular Riemannian metric adapted to T , we prove a uniqueness theorem when the valence of the irregular vertex is not a multiple of 6. Moreover, we exhibit non isomorphic triangulations on X with the same boundary, and with a unique irregular vertex whose valence is 6k, for some k > 1.