Abstract
As we have seen in Chap. 1, a Euclidean triangulated surface \((T_l,M)\) characterizes a polyhedral metric with conical singularities associated with the vertices of the triangulation.
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Notes
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If we glue directly the flat stripes \(z(h,j)^+,\; z(j,k)^+,\) and \(z(k,h)^+\) by suitably identifying their extremities then we would get the familiar conical singularity of \(3\pi\) associated with a zero of order 1 of a J–S quadratic differential. Recall that a zero of order k generates a conical singularity given by \(\pi (k+2).\) This conical singularity is rather annoying since it is not directly related with the conical singularities of the polyhedral surface \((T_l,M).\) It can be eliminated by introducing, at the vertex \(\rho^0 (h,j,k),\) the uniformizing coordinate \(\zeta (h,j,k)\) with \(\zeta (h,j,k)=[z(k,h)^+]^{2/3}.\) This can be seen as another manifestation of Troyanov’s basic observation that conical singularities are invisible from the conformal viewpoint.
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Carfora, M., Marzuoli, A. (2012). Singular Euclidean Structures and Riemann Surfaces. In: Quantum Triangulations. Lecture Notes in Physics, vol 845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24440-7_2
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