Combinatorial Calabi flows on surfaces

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of the smooth Calabi flow. We prove that the solution to the combinatorial Calabi flow exists for all time and converges if and only if the Thurston's circle packing exists. As a consequence, the combinatorial Calabi flow provides a new algorithm to find circle packings with prescribed curvatures. The proofs rely on careful analysis of the combinatorial Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.