离散曲率流若干问题的研究

The study of geometric flow is a very popular topic in geometric analysis, which has a great effect on the development of geometric analysis. Discrete differential geometry is an active field, which combines differential geometry with discrete geometry. The study of discrete curvature flow has played an important role in medical treatment, computer graphics and so on. In this project, we will study the PL-discrete Yamabe flow under hyperbolic geometry background and the combinatorial Yamabe flow under sphere packing background. We hope that under certain conditions, the solution can be extended so that it has longtime existence, and converges to constant curvature metric. We will also study the prescribed discrete Yamabe flow under these two backgrounds corresponding to the prescribing curvature problem. We hope there will be corresponding convergence results.

几何流的研究是几何分析领域的热门课题，对几何分析的发展产生了巨大的推动作用。离散微分几何是一个活跃的领域，它将微分几何与离散几何结合在一起。离散曲率流的研究已经在医疗，计算机图形学等领域发挥了重要的作用。本项目计划研究双曲几何背景下的PL-离散Yamabe流以及球填充背景下的的组合Yamabe流，希望可以证明在一定的条件下，它们的解可以延拓使得延拓之后的解具有长时间存在性，并且收敛到常曲率度量。我们还将研究这两种背景下的预定曲率问题对应的预定曲率离散Yamabe流，希望可以有对应的收敛性结果。