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.

Abstract:
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some 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.

Abstract:
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 $2\pi$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.

Abstract:
In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which could be taken as the area of the hyperbolic disk packed at each vertex. We prove that the corresponding discrete Ricci flow converges if and only if there exists a circle packing metric with zero curvature. We also prove that the flow converges if the initial curvatures are all negative. Note that, this result does not require the existence of zero curvature metric or Thurston's combinatorial-topological condition. We further generalize the definition of combinatorial curvature to any given area element and prove the equivalence between the existence of zero curvature metric and the convergence of the corresponding flow.

Abstract:
In this paper, we introduce two discrete curvature flows, which are called $\alpha$-flows on two and three dimensional triangulated manifolds. For triangulated surface $M$, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming at evolving $\alpha$ order discrete Gauss curvature to a constant. When $\alpha\chi(M)\leq0$, we prove that the convergence of the flow is equivalent to the existence of constant $\alpha$-curvature metric. We further get a necessary and sufficient combinatorial-topological-metric condition, which is a generalization of Thurston's combinatorial-topological condition, for the existence of constant $\alpha$-curvature metric. For triangulated 3-manifolds, we generalize the combinatorial Yamabe functional and combinatorial Yamabe problem introduced by the authors in \cite{GX2,GX4} to $\alpha$-order. We also study the $\alpha$-order flow carefully, aiming at evolving $\alpha$ order combinatorial scalar curvature, which is a generalization of Cooper and Rivin's combinatorial scalar curvature, to a constant.

Abstract:
In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem.

Abstract:
We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.

Abstract:
In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more reasonable definition of discrete Einstein metric than our former version in \cite{G}. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics.

Abstract:
The main utilization of biological resources from coastal wetlands in China and the major limiting factors on sustainable utilization of these resources are reviewed. Strategies for the sustainable utilization of coastal wetlands are proposed. These include further studies on saline agriculture and wetland ecosystems, optimizing methods for maximizing metabolic production from organisms, developing microbial resources and efficient utilization of biological genetic resources, and exploiting bioenergy from coastal wetlands.