Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds

Consider a manifold constructed by identifying the boundaries of Euclidean triangles or Euclidean tetrahedra. When these form a closed topological manifold, we call such spaces piecewise flat manifolds (see Definition 1) as in [8]. Such spaces may be considered discrete analogues of Riemannian manifolds, in that their geometry can be described locally by a finite number of parameters, and the study of curvature on such spaces goes back at least to Regge [35]. In this paper, we give a definition of conformal variation of piecewise flat manifolds in order to study the curvature of such spaces. Conformal variations of Riemannian manifolds have been well studied. While the most general variation formulas for curvature quantities is often complicated, the same formulas under conformal variations often take a simpler form. For this reason, it has even been suggested that an approach to finding Einstein manifolds would be to first optimize within a conformal class, finding a minimum of the Einstein-Hilbert functional within that conformal class, and then maximize across conformal classes to find a critical point of the functional in general [1]. Finding critical points of the Einstein-Hilbert functional within a conformal class is a well-studied problem dating back to Yamabe [48], and the proof that there exists a constant scalar curvature metric in every conformal class was completed by Trudinger [45], Aubin [2], and Schoen [40] (see also [27] for an overview of the Yamabe problem). Implicitly, there has been much work on conformal parametrization of twodimensional piecewise flat manifolds, many of which start with a circle packing on a region in R or a generalized circle packing on a manifold. Thurston found a variational proof of Andreev’s theorem ([44] [29]) and conjectured that the Riemann mapping theorem could be approximated by circle packing maps,