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.
In this paper, we define a new discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study the corresponding constant curvature problem, which is called the combinatorial Yamabe problem, by the corresponding combinatorial versions of Ricci flow and Calabi flow for surfaces and Yamabe flow for 3-dimensional manifolds. The basic tools are the discrete maximal principle and variational principle.
Given a triangulated surface $M$, we use Ge-Xu's $\alpha$-flow \cite{Ge-Xu1}
to deform any initial inversive distance circle packing metric to a metric with
constant $\alpha$-curvature. More precisely, we prove that the inversive
distance circle packing with constant $\alpha$-curvature is unique if
$\alpha\chi(M)\leq 0$, which generalize Andreev-Thurston's rigidity results for
circle packing with constant cone angles. We further prove that the solution to
Ge-Xu's $\alpha$-flow can always be extended to a solution that exists for all
time and converges exponentially fast to constant $\alpha$-curvature. Finally,
we give some combinatorial and topological obstacles for the existence of
constant $\alpha$-curvature metrics.Comment: 14 pages, all comments are welcom
For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than 2π, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.
We define Discrete Quasi-Einstein metrics (DQE-metrics) as critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when the initial energy is small enough. The proof relies on a careful analysis of discrete dual-Laplacians which we interpret as the Jacobian matrix of the curvature map. As a consequence, combinatorial curvature flow provides an algorithm to compute discrete sphere packing metrics with prescribed curvatures.
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