Combinatorial Calabi flows on surfaces

Abstract

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

17 pages, 5 figures

References in corpus (6)

• Discrete Exterior Calculus
• Geometric triangulations and discrete Laplacians on manifolds
• Discrete Quasi-Einstein Metrics and Combinatorial Curvature Flows in 3-Dimension
• Discrete Laplace-Beltrami Operator Determines Discrete Riemannian Metric
• Calabi flow in Riemann surfaces revisited: A new point of view
• Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds

Cited by in corpus (2)

• Discrete Quasi-Einstein Metrics and Combinatorial Curvature Flows in 3-Dimension
• A combinatorial Yamabe problem on two and three dimensional manifolds