Combinatorial Calabi flows on infinitely triangulated surfaces

We study the combinatorial Calabi flow associated with vertex scalings on infinitely triangulated surfaces with piecewise Euclidean and piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an $\epsilon$-uniformly nondegenerate initial metric, we establish short-time existence of smooth solutions and prove uniqueness among solutions that are continuous in the $L^{\mathrm{\infty }}$ norm.