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In [7], a notion of constant scalar curvature metrics on piecewise flatmanifolds is defined. Such metrics are candidates for canonical metrics ondiscrete manifolds. In this paper, we define a class of vertex transitivemetrics on certain triangulations of $\mathbb{S}^3$; namely, the boundarycomplexes of cyclic polytopes. We use combinatorial properties of cyclicpolytopes to show that, for any number of vertices, these metrics have constantscalar curvature.

Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes